The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A320387 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A320387

Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.
(history; published version)

#53 by Michael De Vlieger at Sun Jan 22 11:35:42 EST 2023

STATUS

reviewed

approved

#52 by Joerg Arndt at Sun Jan 22 11:06:10 EST 2023

STATUS

proposed

reviewed

#51 by Andrew Howroyd at Sun Jan 22 11:00:47 EST 2023

STATUS

editing

proposed

#50 by Andrew Howroyd at Sun Jan 22 10:59:33 EST 2023

FORMULA

G.f.: Sum_{k>=1} x^binomial(k,2)/Product_{j=1..k-1} (1 - x^(binomial(k,2)-binomial(j,2))). - Andrew Howroyd, Jan 22 2023

PROG

(PARI) seq(n)={Vec(sum(k=1, (sqrtint(8*n+1)+1)\2, my(t=binomial(k, 2)); x^t/prod(j=1, k-1, 1 - x^(t-binomial(j, 2)) + O(x^(n-t+1)))))} \\ Andrew Howroyd, Jan 22 2023

STATUS

approved

editing

#49 by Michael De Vlieger at Tue Jan 17 21:25:00 EST 2023

STATUS

proposed

approved

#48 by Gus Wiseman at Tue Jan 17 18:50:37 EST 2023

STATUS

editing

proposed

#47 by Gus Wiseman at Tue Jan 17 18:50:01 EST 2023

CROSSREFS

A359497 gives max for given weighted Weighted sum of prime indices, zero-based A359757: A359497, A359676, A359682, A359754, A359755.

A359682 gives min for given weighted sum of prime indices, zero-based A359676.

A359755 gives first positions for weighted sums of prime indices, rev A359754.

#46 by Gus Wiseman at Tue Jan 17 16:07:35 EST 2023

CROSSREFS

A358194 counts partitions by weighted sum, reverse A264034.

Cf. A029931, A243055, A264034, `A325362, A358136, A358137, A358194, `A359361, A359397, A359674, A359677, A359678, A359681.

#45 by Gus Wiseman at Tue Jan 17 15:53:45 EST 2023

COMMENTS

Also Equivalently, a(n) is the number of multisets (weakly increasing sequences of positive integers whose prime indices have ) with weighted sum n. For example, the Heinz numbers of the a(0) = 1 through a(15) = 7 numbers multisets are:

These multisets are ranked counted by A264034. The reverse version is A007294. The zero-based version is A359678.

#44 by Gus Wiseman at Tue Jan 17 14:35:13 EST 2023

CROSSREFS

A359497 gives maximum max for given weighted sum of prime indices, zero-based A359757.

A359682 gives minimum min for given weighted sum of prime indices, zero-based A359676 (reverse A359681).

A359755 gives first positions for weighted sums of prime indices, reverse rev A359754.

`Cf. A000009, A029931, A243055, A264034, `A325362, ~`A355536, ~`A358133, A358136, A358137, A358194, `A359361, A359397, A359674, A359675, A359677, A359678, `A359680A359681.