THE FOLD / LOOT / THE BOUNTY / THE DOBINSKI
THE DOBINSKI
an infinite series that lands on an integer
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Dobiński’s formula writes a whole number as an infinite series. The Bell number Bn counts the ways to partition a set of n elements into non-empty blocks — a pure combinatorial integer (1, 1, 2, 5, 15, 52, 203, …). Dobiński’s formula says this integer equals an infinite sum divided by e: Bn = (1/e)∑k≥0 kn/k!. Each term kn/k! is irrational, e is transcendental, yet the whole thing lands exactly on an integer — a Poisson-distribution moment in disguise.
LIT verified live: for n = 0..13, the truncated series (1/e)∑ kn/k! rounds to exactly the Bell number computed independently by the Bell triangle recurrence, with relative error ~1e-15 (window.__dobinski). FIG no framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree.
LIT verified live: for n = 0..13, the truncated series (1/e)∑ kn/k! rounds to exactly the Bell number computed independently by the Bell triangle recurrence, with relative error ~1e-15 (window.__dobinski). FIG no framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the payout: an infinite series of irrational terms, scaled by 1/e, pays out an exact whole-number count. AVAN (AI) built the instrument: the Dobiński series, the independent Bell-triangle recurrence, and their agreement.
Credit as content: G. Dobiński (1877). The weave: David names the payout; I confirm the transcendental series lands exactly on the Bell number.
Credit as content: G. Dobiński (1877). The weave: David names the payout; I confirm the transcendental series lands exactly on the Bell number.
3 ONE DIMENSION
The Dobiński terms k^n/k! (they peak near k=n) whose sum, divided by e, is exactly the Bell number.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the truncated (1/e)Σ k^n/k! is compared to the Bell number from the recurrence.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Bell number, an exact integer.
AVAN’s addition (the inverse-companion): don’t count partitions — sum a Poisson series. The inverse of ‘the integer Bn’ is ‘the infinite sum (1/e)∑kn/k! whose irrational terms cancel to it’ — it is the n-th moment of a Poisson(1) variable. Magenta are the infinite series terms; green is the integer they sum to. A whole number wearing an infinite series.
LIT Genuine Dobiński's formula (G. Dobiński, 1877). Verified live: for n=0..13 the truncated series (1/e)Σ_{k≥0}k^n/k! rounds to exactly the Bell number B_n computed independently by the Bell-triangle recurrence, worst relative error ~1.3e-15 (window.__dobinski.ok, .worst).
FIG No framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree. The AVAN inverse is honest — instead of counting partitions, sum a Poisson series: the inverse of 'the integer B_n' is 'the infinite sum (1/e)Σk^n/k! whose irrational terms cancel to it', the n-th moment of a Poisson(1) variable. Magenta are the infinite series terms; green is the integer they sum to. A whole number wearing an infinite series.
FIG No framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree. The AVAN inverse is honest — instead of counting partitions, sum a Poisson series: the inverse of 'the integer B_n' is 'the infinite sum (1/e)Σk^n/k! whose irrational terms cancel to it', the n-th moment of a Poisson(1) variable. Magenta are the infinite series terms; green is the integer they sum to. A whole number wearing an infinite series.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN