cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A307260 Expansion of (1/(1 + x)) * Product_{k>=1} (1 + k*x^k/(1 + x)^k).

Original entry on oeis.org

1, 0, 1, 1, -4, 14, -35, 77, -161, 356, -873, 2267, -5787, 13850, -30361, 59934, -103754, 147968, -139049, -58998, 730972, -2430881, 6333238, -15548722, 39845197, -110775861, 325257904, -960503811, 2756222486, -7568564555, 19815541729, -49548068461, 118752506024
Offset: 0

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Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Inverse binomial transform of A022629.

Crossrefs

Cf. A022629, A293467, A294503, A307258, A307259.

Programs

  • Maple
    a:=series((1/(1+x))*mul(1+k*x^k/(1+x)^k,k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 + x) Product[(1 + k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A022629(k).

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280
Offset: 0

Views

Author

Gus Wiseman, Jan 12 2025

Keywords

Comments

Also the inverse zero-based binomial transform of the odd prime numbers.

Crossrefs

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

Programs

  • Mathematica
    nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

A304781 a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 6, 21, 75, 274, 1016, 3807, 14377, 54627, 208584, 799669, 3076167, 11867511, 45897145, 177888715, 690770763, 2686879415, 10466761637, 40828165464, 159453481037, 623427464093, 2439907421914, 9557831470082, 37472409664888, 147028505564603, 577302980976146
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Number of partitions of n into odd parts with n + 1 kinds of 1.

Links

Crossrefs

Cf. A000009, A036469, A095944, A128566, A128593, A292463, A292613, A293467.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x)^n Product[(1 + x^k), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Product[1/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[QPochhammer[-1, x]/(2 (1 - x)^n), {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} 1/(1 - x^(2*k-1)).
a(n) = [x^n] (1/(1 - x)^n)*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) ~ QPochhammer[-1, 1/2] * 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, May 18 2018
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