A294254 -id:A294254 - cp's OEIS frontend

A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 5, 13, 0, 1, 1, 5, 25, 73, 0, 1, 1, 5, 31, 193, 501, 0, 1, 1, 5, 31, 241, 1601, 4051, 0, 1, 1, 5, 31, 265, 2261, 16741, 37633, 0, 1, 1, 5, 31, 265, 2501, 25501, 190345, 394353, 0, 1, 1, 5, 31, 265, 2621, 29461, 319915, 2509025

Offset: 0

Seiichi Manyama, Oct 25 2017

			Square array B(j,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   1,    2,    2,    2, ...
   0,   1,    2,    3,    3, ...
   0,   1,    3,    4,    5, ...
   0,   1,    3,    5,    6, ...
Square array A(n,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   3,    5,    5,    5, ...
   0,  13,   25,   31,   31, ...
   0,  73,  193,  241,  265, ...
   0, 501, 1601, 2261, 2501, ...

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

Columns k=0..5 give A000007, A000262, A294213, A294214, A294215, A294216.
Rows n=0 gives A000012.
Main diagonal gives A058892.
Cf. A058398, A294250, A294254, A294289.

B(j,k) is the coefficient of Product_{i=1..k} 1/(1-x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 13, 1, 0, 1, 1, 3, 19, 49, 1, 0, 1, 1, 3, 19, 97, 261, 1, 0, 1, 1, 3, 19, 121, 681, 1531, 1, 0, 1, 1, 3, 19, 121, 921, 5971, 9073, 1, 0, 1, 1, 3, 19, 121, 1041, 8491, 50443, 63393, 1, 0, 1, 1, 3, 19, 121, 1041

Offset: 0

Seiichi Manyama, Oct 26 2017

			Square array A(n,k) begins:
   1, 1,   1,   1,   1, ...
   0, 1,   1,   1,   1, ...
   0, 1,   3,   3,   3, ...
   0, 1,  13,  19,  19, ...
   0, 1,  49,  97, 121, ...
   0, 1, 261, 681, 921, ...

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

Columns k=0..5 give A000007, A000012, A118589, A294251, A294252, A294253.
Rows n=0 gives A000012.
Main diagonal gives A293840.
Cf. A070936, A294212, A294254.

B(j,k) is the coefficient of Product_{i=1..k} (1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A294260 E.g.f.: exp(Sum_{n>=1} A010815(n)*x^n).

Original entry on oeis.org

1, -1, -1, 5, 1, 79, -689, 2981, -7615, -172801, 3621151, -16469531, -240199871, 2722511375, 51840080111, -1987808959291, 12337235928961, 136594696115071, -1167414675803585, -56631124939839931, -1376838916423621759, 69766591820556094799

Offset: 0

Seiichi Manyama, Oct 26 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..449

Main diagonal of A294254.

Cf. A293840.

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A010815(k)*a(n-k)/(n-k)! for n > 0.

A294255 E.g.f.: exp((1-x)*(1-x^2) - 1).

Original entry on oeis.org

1, -1, -1, 11, -23, -101, 991, -1849, -24751, 220823, -174689, -10924541, 87639289, 105372851, -7496679553, 52397291519, 238887991201, -7313210597969, 41947208776639, 440627548672763, -9537975555701239, 39732691614159179, 916052993052974239

Offset: 0

Seiichi Manyama, Oct 26 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..578

Column k=2 of A294254.

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

A294256 E.g.f.: exp((1-x)(1-x^2)(1-x^3) - 1).

Original entry on oeis.org

1, -1, -1, 5, 25, -41, -1769, 7181, 74705, -236305, -7562609, -4169771, 1339186729, -2144219065, -174083994265, -370106105251, 35885107199521, 252732128552671, -10892824509622625, -56149731134394715, 2589936417283432121, 31565855231279227319

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..491

Column k=3 of A294254.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1-x)*(1-x^2)*(1-x^3)-1)))

A294257 E.g.f.: exp((1-x)(1-x^2)(1-x^3)*(1-x^4) - 1).

Original entry on oeis.org

1, -1, -1, 5, 1, 199, -1409, -4579, 25985, 23759, 13872511, -129511691, -1050605951, 4703204375, 33833371391, 6781977225869, -71864943455999, -926469142419809, -2738986440688385, 11032808190085349, 16738876153047304961, -146356319119622810521

Offset: 0

Seiichi Manyama, Oct 26 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..467

Column k=4 of A294254.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1-x)*(1-x^2)*(1-x^3)*(1-x^4)-1)))

A294258 Expansion of e.g.f.: exp((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5) - 1).

Original entry on oeis.org

1, -1, -1, 5, 1, 79, 31, -2059, -68095, 129599, 5586751, 36420229, -159700991, -6256012465, -94786800289, -3320395014091, 74018996229121, 1093450523140351, -4548151329265025, -222875601233847931, -7118941159301980159, -60758029090422335281

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..458

Column k=5 of A294254.

With[{nn=30},CoefficientList[Series[Exp[Times@@(1-x^Range[5])-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 29 2020 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)-1)))

Definition clarified by Harvey P. Dale, Oct 29 2020

cp's OEIS Frontend

A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

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A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

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A294260 E.g.f.: exp(Sum_{n>=1} A010815(n)*x^n).

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A294255 E.g.f.: exp((1-x)*(1-x^2) - 1).

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PARI

A294256 E.g.f.: exp((1-x)*(1-x^2)*(1-x^3) - 1).

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PARI

A294257 E.g.f.: exp((1-x)*(1-x^2)*(1-x^3)*(1-x^4) - 1).

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PARI

A294258 Expansion of e.g.f.: exp((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5) - 1).

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Extensions

A294256 E.g.f.: exp((1-x)(1-x^2)(1-x^3) - 1).

A294257 E.g.f.: exp((1-x)(1-x^2)(1-x^3)*(1-x^4) - 1).

A294258 Expansion of e.g.f.: exp((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5) - 1).