A024427 -id:A024427 - cp's OEIS frontend

A171367 Antidiagonal sums of triangle of Stirling numbers of 2nd kind A048993.

1, 0, 1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, 5379, 19195, 71872, 281571, 1151338, 4902687, 21696505, 99598840, 473466698, 2327173489, 11810472444, 61808852380, 333170844940, 1847741027555, 10532499571707, 61649191750137, 370208647200165, 2278936037262610, 14369780182166215

Offset: 0

Paul Barry, Dec 06 2009

Seiichi Manyama, Table of n, a(n) for n = 0..665
P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics, Volume 32, Issue 2, 1980, pp. 125-161.

Cf. A024427, A097341, A097342.

b:= proc(n, m) option remember; `if`(n<=m,
     `if`(n=m, 1, 0), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2023

Table[Sum[StirlingS2[n-k, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 18 2016 *)

makelist(sum(stirling2(n-k,k),k,0,n),n,0,60); /* Emanuele Munarini, Jun 01 2012 */

a(n) = sum(k=0, n, stirling(n-k, k,2)); /* Joerg Arndt, Jan 16 2013 */

G.f.: 1/(1-x^2/(1-x/(1-x^2/(1-2x/(1-x^2/1-3x/(1-x^2/(1-4x/(1-x^2/(1-5x/(1-... (continued fraction).
G.f.: (G(0) - 1)/(x-1)/x where G(k) =  1 - x/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: T(0)/(1-x^2), where T(k) = 1-x^3*(k+1)/(x^3*(k+1)-(1-x*(x+k))*(1-x*(x+1+k))/T(k+1) ); (continued fraction, after P. Flajolet, p. 140). - Sergei N. Gladkovskii, Oct 30 2013
G.f. (alternating signs): Sum_{k>=0} S(x,k)*x^k, where S(x,k)*exp(-x) is the inverse Mellin transform of Gamma(s)*s^k. - Benedict W. J. Irwin, Oct 14 2016

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

Original entry on oeis.org

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

A097342 a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 3^k.

Original entry on oeis.org

1, 0, 3, 3, 12, 30, 93, 300, 1038, 3810, 14781, 60375, 258807, 1160949, 5435652, 26502555, 134282406, 705720549, 3840542031, 21608662710, 125523530724, 751831408929, 4637611255422, 29428408797852, 191907975348210

Offset: 0

Paul Barry, Aug 05 2004

Seiichi Manyama, Table of n, a(n) for n = 0..647

Cf. A024427, A097341.

Table[Sum[StirlingS2[n-k,k]3^k,{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, May 24 2012 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, 3^k*x^(2*k)/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Apr 09 2022

a(n) = sum(k=0, n\2, 3^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)!)}3^k }

G.f.: Sum_{k>=0} 3^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

A119429 Expansion of Sum_{k>=0} x^(2k)/Product_{j=1..k} (1 - j2*x).

Original entry on oeis.org

1, 0, 1, 2, 5, 14, 45, 164, 661, 2884, 13461, 66894, 353217, 1977146, 11691481, 72734088, 474172777, 3229062120, 22914397417, 169128976922, 1296276808253, 10300677006854, 84731125615749, 720392483485868, 6321631421441149

Offset: 0

Paul Barry, May 19 2006

Seiichi Manyama, Table of n, a(n) for n = 0..588

Cf. A024427, A048993.

a(n) = sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)); \\ Seiichi Manyama, Apr 08 2022

a(n) = Sum_{k=0..n} S2(k,n-k)*2^(2k-n) where S2(n,k)=A048993(n,k).
a(n) = Sum_{k=0..floor(n/2)} S2(n-k,k)*2^(n-2k) where S2(n,k)=A048993(n,k).
Let E(x) = Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - j*2*x), then E(x) = 1 + x^2/(U(0)-x^2), where U(k) = (x-1)^2 - 2*k*x + x^2*(2*k*x + 2*x - 1)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2012
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - x/(1-x*(2*k+2))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

A136011 Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 6, 1, 31, 25, 1, 1, 63, 90, 10, 1, 127, 301, 65, 1, 1, 255, 966, 350, 15, 1, 511, 3025, 1701, 140, 1, 1, 1023, 9330, 7770, 1050, 21, 1, 2047, 28501, 34105, 6951, 266, 1, 1, 4095, 86526, 145750, 42525, 2646, 28

Offset: 1

Gary W. Adamson, Dec 09 2007

Row sums = A024427: (1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, ...).

T(n,k) is the number of ways to partition {1,2,...,n+1} into exactly k blocks such that each block contains at least 2 elements and the smallest 2 elements in each block are consecutive integers. - Geoffrey Critzer, Dec 02 2013

			First few rows of the triangle:
  1;
  1;
  1,   1;
  1,   3;
  1,   7,   1;
  1,  15,   6;
  1,  31,  25,   1;
  1,  63,  90,  10;
  1, 127, 301,  65,  1;
  1, 255, 966, 350, 15;
  ...
T(5,3) = 1 because we have {1,2},{3,4},{5,6}.
T(6,3) = 6 because we have {1,2,7},{3,4},{5,6}; {1,2},{3,4,7},{5,6}; {1,2},{3,4},{5,6,7}; {1,2},{3,4,5},{6,7}; {1,2,3},{4,5},{6,7}; {1,2,5},{3,4},{6,7}. - _Geoffrey Critzer_, Dec 02 2013

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A008277, A024427.

T:= (n, k)-> Stirling2(n+1-k, k):
seq(seq(T(n, k), k=1..(n+1)/2), n=1..20); # Alois P. Heinz, Dec 04 2013

nn=15;Range[0,nn]!;Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[Product[x^2/(1-i x),{i,1,k}],{x,0,nn}],x],{k,1,nn/2}]],2]]//Grid (* Geoffrey Critzer, Dec 02 2013 *)

Given A008277, the Stirling number of the second kind triangle, left column = (1, 1, 1, ...); all other columns start at 3rd term of previous column.
O.g.f. for column k: Product_{i=1..k} x^2/(1 - i*x). - Geoffrey Critzer, Dec 02 2013
T(n,k) = Stirling2(n+1-k,k). - Alois P. Heinz, Dec 04 2013

A137856 Row sums of triangle A137855.

Original entry on oeis.org

1, 2, 4, 8, 17, 39, 97, 261, 756, 2343, 7722, 26917, 98789, 380360, 1531698, 6434385, 28130890, 127729730, 601196428, 2928369917, 14738842361, 76547694741, 409718539681, 2257459567236, 12789959138943, 74439150889080, 444647798089245, 2723583835351855

Offset: 1

Gary W. Adamson, Feb 18 2008

nonn

			a(5) = 17 = sum of row 5 terms of triangle A137855: (1 + 2 + 5 + 8 + 1).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A137855.

Partial sums of A024427.

a(n)={sum(k=1, n, sum(j=1, n-k+1, stirling(k,j,2)))} \\ Andrew Howroyd, Aug 09 2018

a(n) = Sum_{k=1..n} Sum_{j=1..n-k+1} Stirling2(k, j). - Andrew Howroyd, Aug 09 2018

Terms a(12) and beyond from Andrew Howroyd, Aug 09 2018

A176599 Numerators of the first column of a table with top row the nonnegative integers and successive rows defined by a reverted Akiyama-Tanigawa procedure.

Original entry on oeis.org

1, -1, -1, -5, -7, -1631, -96547, -40291823, -16870575007, -7075000252463, -2969301738826267, -13713149169712887583, -10557203537780702505907

Offset: 0

Paul Curtz, Apr 21 2010

Define a reverted Akiyama-Tanigawa procedure which takes a sequence s(1), s(2), s(3), ..., as input and constructs the sequence of (s(k)-s(k+1))/k as output. (The difference from the standard algorithm is that the differences are divided by k, not multiplied by k.)
Starting from a top row with nonnegative integers, the following table is constructed row after row by applying the reverted algorithm in succession:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
-1, -1/2, -1/3, -1/4, -1/5, -1/6, -1/7, -1/8, -1/9, -1/10, -1/11, ...
-1/2, -1/12, -1/36, -1/80, -1/150, -1/252, -1/392, -1/576, -1/810, ...
-5/12, -1/36, -11/2160, -7/4800, -17/31500, -5/21168, -23/197568, ...
-7/18, -49/4320, -157/129600, -463/2016000, -803/13230000, ...
-1631/4320, -1313/259200, -17813/54432000, -35767/846720000, ...
-96547/259200, -257917/108864000, -2171917/22861440000, ...
The numerators of the left column define the current sequence.
The denominators of the third row are in A011379.

			From _Peter Bala_, Aug 14 2012: (Start)
Column 2: Sum_{n >= 2} x^(n-1)/(Product_{k = 2..n} (x-k)) = -(1/2)*x - (1/12)*x^2 - (1/36)*x^3 - (49/4320)*x^4 - ...
Column 3: Sum_{n >= 3} x^(n-2)/(Product_{k = 3..n} (x-k)) = -(1/3)*x - (1/36)*x^2 - (11/2160)*x^3 - (157/129600)*x^4 - .... (End)

Cormac O'Sullivan, Stirling's approximation and a hidden link between two of Ramanujan's approximations, arXiv:2208.02898 [math.NT], 2022. See p. 17.
Craig A. Tracy and H. Widom, On the ground state energy of the delta-function Bose gas, arXiv preprint arXiv:1601.04677 [math-ph], 2016.

Cf. A024427.

a[1, k_] := k; a[n_, k_] := a[n, k] = (a[n-1, k] - a[n-1, k+1])/k; a[n_] := Numerator[a[n, 1]]; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Aug 02 2012 *)

From Peter Bala, Aug 14 2012: (Start)
The o.g.f. for the rational numbers in the first column of the above table is Sum_{n >= 0} x^n/(Product_{k = 1..n} (x-k)) = 1 - x - 1/2*x^2 - 5/12*x^3 - 7/18*x^4 - .... This yields the formula |a(n)| = numerator of Sum_{k = 0..n-1} (1/k!) * Sum_{i = 0..k} (-1)^i*binomial(k,i)*(k-i+1)^(k-n). Cf. A024427.
More generally, the o.g.f. for the rational numbers in the r-th column of the above table (excluding the first entry of r) is Sum_{n >= r} x^(n+1-r)/ (Product_{k = r..n} (x-k)).
(End)
The first column of the above table lists the coefficients of the expansion of b(1)x/(1+b(2)x/(1+b(3)x/(1+b(4)x/(...)))), a continued fraction, where b(n) are -1, -1/2, -1/3, -1/4, ... i.e., the second row of the table above. - Benedict W. J. Irwin, May 10 2016

cp's OEIS Frontend

A171367 Antidiagonal sums of triangle of Stirling numbers of 2nd kind A048993.

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

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A097342 a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 3^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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A119429 Expansion of Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - j*2*x).

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A136011 Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.

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A137856 Row sums of triangle A137855.

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A176599 Numerators of the first column of a table with top row the nonnegative integers and successive rows defined by a reverted Akiyama-Tanigawa procedure.

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

A119429 Expansion of Sum_{k>=0} x^(2k)/Product_{j=1..k} (1 - j2*x).