A292894 -id:A292894 - cp's OEIS frontend

A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570

Offset: 0

Seiichi Manyama, Sep 26 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   1,  0,  0,  0,   0, ...
   2,  2,  0,  0,   0, ...
   5,  3,  6,  0,   0, ...
  15, 16, 12, 24,   0, ...
  52, 65, 20, 60, 120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000110, A052506, A240989, A292891.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A292894, A355607.

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A292893 Expansion of e.g.f. exp(x * (1 - exp(x))).

Original entry on oeis.org

1, 0, -2, -3, 8, 55, 84, -637, -4992, -10593, 92060, 1012099, 3642000, -18733585, -354606084, -2157876645, 2003383424, 175455790399, 1766183783868, 5436448194707, -96997103373360, -1770215099996721, -13073420293290148, 22275369715313131

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..556

Column k=1 of A292894.

Cf. A000587, A052506, A080108.

x='x+O('x^66); Vec(serlaplace(exp(x*(1-exp(x)))))

a(n) = n!*sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=2, i, j/(j-1)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) = sum(k=0, n, (-1)^k*(k+1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * Stirling2(n-k,k)/(n-k)!.
a(0) = 1; a(n) = -(n-1)! * Sum_{k=2..n} k/(k-1)! * a(n-k)/(n-k)!. (End)
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k * (k+1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} (-x)^k / (1 - (k+1)*x)^(k+1). (End)

A292951 Expansion of e.g.f. exp(x^2 * (1 - exp(x))).

Original entry on oeis.org

1, 0, 0, -6, -12, -20, 330, 2478, 11704, -15192, -751050, -7817150, -38408172, 151402524, 5793891922, 69046056870, 393083614320, -2517944476592, -98819987200146, -1384209703077750, -9376308260215220, 67288368700200900, 3186749671049174538

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200

Column k=2 of A292894.

Cf. A240989.

With[{nn=30},CoefficientList[Series[Exp[x^2 (1-Exp[x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 16 2021 *)

x='x+O('x^66); Vec(serlaplace(exp(x^2*(1-exp(x)))))

a(n) = n!*sum(k=0, n\3, (-1)^k*stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = -(n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

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A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

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A292893 Expansion of e.g.f. exp(x * (1 - exp(x))).

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A292951 Expansion of e.g.f. exp(x^2 * (1 - exp(x))).

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