A171367 -id:A171367 - cp's OEIS frontend

A097341 a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 2^k.

1, 0, 2, 2, 6, 14, 38, 110, 342, 1134, 3990, 14830, 58006, 237998, 1021462, 4574318, 21325462, 103287598, 518768406, 2697426926, 14498316182, 80440333998, 460112203798, 2710038058862, 16418576767126, 102212840258094, 653247225514262, 4282249051881198

Offset: 0

Paul Barry, Aug 05 2004

Seiichi Manyama, Table of n, a(n) for n = 0..654

Cf. A024427, A097342, A119429, A119430, A171367.

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, 2^k*x^(2*k)/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Apr 09 2022

a(n) = sum(k=0, n\2, 2^k*stirling(n-k, k, 2)); \\ Seiichi Manyama, Apr 09 2022

a(n)=sum{k=0..floor(n/2), sum{i=0..k, (-1)^(k+i)i^(n-k)/(i!(k-i)!)}2^k }

G.f.: Sum_{k>=0} 2^k * x^(2*k)/Product_{j=1..k} (1 - j * x). - Seiichi Manyama, Apr 09 2022

A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 57, 157, 453, 1368, 4296, 13995, 47138, 163779, 585741, 2152349, 8113188, 31326760, 123748871, 499539900, 2058542819, 8651755865, 37054078481, 161591063250, 717032333816, 3235298221401, 14834735654080, 69085973044125

Offset: 0

Paul D. Hanna, Oct 28 2006

nonn

The Kn11 triangle sums of A094638 are given by the terms of this sequence. For the definitions of this and other triangle sums see A180662. [Johannes W. Meijer, Apr 20 2011]

			A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..868
Giulio Cerbai, Anders Claesson, and Bruce E. Sagan, Self-modified difference ascent sequences, arXiv:2408.06959 [math.CO], 2024. See p. 15.

Cf. A024427, A171367.

nmax = 30; CoefficientList[Series[Sum[x^(2*k)*Pochhammer[1 + 1/x, k], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2024 *)
Table[Sum[(-1)^k * StirlingS1[n+1-k, n+1-2*k], {k, 0, (n+1)/2}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 18 2024 *)

a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+j*x+x*O(x^n))),n)

O.g.f.: A(x) = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1+x*(k*x+x+1) - x*(k*x + 2*x + 1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/(x*Q(0)-1)/x^4 + (1+x-x^3)/x^4, where Q(k)= 1 - x/(1 - (k+1)*x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: log(a(n)) ~ n*log(n)/2 - n*(1 + log(2))/2. - Vaclav Kotesovec, Sep 18 2024

A320964 a(n) = Sum_{j=0..n} Sum_{k=0..j} Stirling2(j - k, k).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 18, 40, 98, 262, 757, 2344, 7723, 26918, 98790, 380361, 1531699, 6434386, 28130891, 127729731, 601196429, 2928369918, 14738842362, 76547694742, 409718539682, 2257459567237, 12789959138944, 74439150889081, 444647798089246, 2723583835351856

Offset: 0

Peter Luschny, Nov 06 2018

nonn

The row sums of A320955 seen as a triangle are the partial sums of the antidiagonal sums of the triangle of the Stirling set numbers.

Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block m-j contains n-j (m in {0..n}, j in {0..m-1}). a(5) = 9: 12345, 1234|5, 123|4|5, 124|35, 12|3|4|5, 134|25, 13|24|5, 14|235, 1|2|3|4|5. - Alois P. Heinz, May 16 2023

Alois P. Heinz, Table of n, a(n) for n = 0..665

Row sums of A320955 seen as a triangle.

Cf. A171367, A048993.

ListTools:-PartialSums([seq(add(Stirling2(n-k, k), k=0..n), n=0..29)]);
# second Maple program:
b:= proc(n, m) option remember; `if`(n>m,
      b(n-1, m)*m+b(n-1, m+1), `if`(n=m, 1, 0))
    end:
a:= proc(n) a(n):= `if`(n=0, 0, a(n-1))+b(n, 0) end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2023

a[n_] := Sum[Sum[StirlingS2[j - k, k], {k, 0, j}], {j, 0, n}]; Array[a, 30, 0] (* Amiram Eldar, Nov 06 2018 *)
Table[Sum[StirlingS2[j-k,k],{j,0,n},{k,0,j}],{n,0,30}] (* Harvey P. Dale, May 15 2019 *)

a(n)={sum(j=0, n, sum(k=0, j, abs(stirling(j-k, k, 2))))} \\ Andrew Howroyd, Nov 06 2018

A353260 Expansion of Sum_{k>=0} (-1)^k * x^(2k)/Product_{j=1..k} (1 - j x).

Original entry on oeis.org

1, 0, -1, -1, 0, 2, 5, 8, 6, -18, -111, -377, -953, -1567, 964, 23411, 133702, 554185, 1801323, 3910514, -2415952, -92788743, -700128734, -3842587204, -17042883146, -57693979779, -86308109341, 779904767601, 10180307035351, 78523141206142, 481780714913151

Offset: 0

Seiichi Manyama, Apr 09 2022

sign

Cf. A097341, A097342, A171367, A353261.

Cf. A353253.

a[n_] := Sum[(-1)^k * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 09 2022 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (-1)^k*x^(2*k)/prod(j=1, k, 1-j*x)))

a(n) = sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2));

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * Stirling2(n-k,k).

A353261 Expansion of Sum_{k>=0} (-2)^k * x^(2k)/Product_{j=1..k} (1 - j x).

Original entry on oeis.org

1, 0, -2, -2, 2, 10, 18, 10, -62, -310, -894, -1590, 642, 21514, 120322, 461130, 1230466, 877194, -16158974, -142301750, -798423166, -3397990646, -9764986878, 2009650762, 294960691330, 2788851766154, 18403159253250, 95083470290634, 350847712602498

Offset: 0

Seiichi Manyama, Apr 09 2022

sign

Cf. A097341, A097342, A171367, A353260.

Cf. A353254.

a[n_] := Sum[(-2)^k * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 09 2022 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (-2)^k*x^(2*k)/prod(j=1, k, 1-j*x)))

a(n) = sum(k=0, n\2, (-2)^k*stirling(n-k, k, 2));

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n-k,k).

A353262 Expansion of Sum_{k>=0} x^(2k)/Product_{j=1..k} (1 - 3 j * x).

Original entry on oeis.org

1, 0, 1, 3, 10, 36, 145, 666, 3466, 19956, 124111, 821601, 5755987, 42634089, 333827776, 2759262897, 24000288202, 218806121205, 2082848200057, 20639203885008, 212441617055458, 2268057343273491, 25085332185250564, 287096974919978292, 3395697093278589844

Offset: 0

Seiichi Manyama, Apr 09 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..549

Cf. A119429, A171367.

Cf. A353256.

a[n_] := Sum[3^(n-2*k) * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(2*k)/prod(j=1, k, 1-3*j*x)))

a(n) = sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2));

a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k).

A357903 a(n) = Sum_{k=0..floor(n/3)} Stirling2(n - 2*k,k).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 4, 8, 17, 38, 89, 219, 567, 1543, 4400, 13094, 40507, 129874, 430731, 1476030, 5222544, 19066758, 71764369, 278166767, 1108986222, 4541765652, 19085377108, 82211094414, 362717859475, 1638071537802, 7567876937002, 35748311794246, 172558399424154

Offset: 0

Seiichi Manyama, Oct 19 2022

nonn

Cf. A000110, A171367, A357904.

a(n) = sum(k=0, n\3, stirling(n-2*k, k, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/prod(j=1, k, 1-j*x)))

G.f.: Sum_{k>=0} x^(3*k)/Product_{j=1..k} (1 - j * x).

A357904 a(n) = Sum_{k=0..floor(n/4)} Stirling2(n - 3*k,k).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 4, 8, 16, 33, 70, 153, 346, 814, 2000, 5138, 13776, 38395, 110695, 328638, 1001306, 3124626, 9978906, 32620854, 109225582, 374875483, 1319392590, 4761630252, 17610041358, 66668257846, 258018795970, 1019440760020, 4106982942054

Offset: 0

Seiichi Manyama, Oct 19 2022

nonn

Cf. A000110, A171367, A357903.

a(n) = sum(k=0, n\4, stirling(n-3*k, k, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(4*k)/prod(j=1, k, 1-j*x)))

G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..k} (1 - j * x).

A353287 a(n) = Sum_{k=0..floor(n/2)} k^k * Stirling2(n-k,k).

Original entry on oeis.org

1, 0, 1, 1, 5, 13, 56, 223, 1056, 5243, 28401, 163578, 1003332, 6506149, 44464510, 319066188, 2396942740, 18800878491, 153611297283, 1304600660023, 11495292868763, 104907727533628, 990067627794487, 9648859125705064, 96978616443859923

Offset: 0

Seiichi Manyama, Apr 09 2022

nonn

Cf. A097341, A097342, A171367, A353260, A353261.

Cf. A353288.

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k^k*x^(2*k)/prod(j=1, k, 1-j*x)))

a(n) = sum(k=0, n\2, k^k*stirling(n-k, k, 2));

G.f.: Sum_{k>=0} k^k * x^(2*k)/Product_{j=1..k} (1 - j * x).

A353288 a(n) = Sum_{k=0..floor(n/2)} k^(n-2k) Stirling2(n-k,k).

Original entry on oeis.org

1, 0, 1, 1, 2, 7, 30, 139, 723, 4487, 33551, 289854, 2774999, 29016343, 333139222, 4232908176, 59442337179, 912948755487, 15154215501815, 269933506466203, 5150440487875190, 105326085645729766, 2307425141636199329, 53998118146846356916, 1343998910355295080556

Offset: 0

Seiichi Manyama, Apr 09 2022

nonn

Cf. A119429, A171367, A353262.

Cf. A104872, A353287.

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(2*k)/prod(j=1, k, 1-k*j*x)))

a(n) = sum(k=0, n\2, k^(n-2*k)*stirling(n-k, k, 2));

G.f.: Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - k * j * x).

cp's OEIS Frontend

A097341 a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 2^k.

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A124380 O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + k*x).

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A320964 a(n) = Sum_{j=0..n} Sum_{k=0..j} Stirling2(j - k, k).

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A353260 Expansion of Sum_{k>=0} (-1)^k * x^(2*k)/Product_{j=1..k} (1 - j * x).

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A353261 Expansion of Sum_{k>=0} (-2)^k * x^(2*k)/Product_{j=1..k} (1 - j * x).

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A353262 Expansion of Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - 3 * j * x).

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Mathematica

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A357903 a(n) = Sum_{k=0..floor(n/3)} Stirling2(n - 2*k,k).

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A357904 a(n) = Sum_{k=0..floor(n/4)} Stirling2(n - 3*k,k).

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A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).

A353260 Expansion of Sum_{k>=0} (-1)^k * x^(2k)/Product_{j=1..k} (1 - j x).

A353261 Expansion of Sum_{k>=0} (-2)^k * x^(2k)/Product_{j=1..k} (1 - j x).

A353262 Expansion of Sum_{k>=0} x^(2k)/Product_{j=1..k} (1 - 3 j * x).

A353288 a(n) = Sum_{k=0..floor(n/2)} k^(n-2k) Stirling2(n-k,k).