A294289 -id:A294289 - 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.

A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

Original entry on oeis.org

1, -1, 1, -7, 49, -301, 2281, -21211, 260737, -3254329, 41086801, -589336111, 9851907121, -170708882917, 3060177746809, -60544788499651, 1298663388032641, -28777111728560881, 665551703689032097, -16413980708818538839, 428253175770218766001

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..437

Main diagonal of A294289.

Cf. A058892, A293840.

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A081362(k)*a(n-k)/(n-k)! for n > 0.

A294291 E.g.f.: exp(1/((1+x)(1+x^2)(1+x^3)) - 1).

Original entry on oeis.org

1, -1, 1, -7, 73, -421, 2641, -32131, 403537, -4527433, 57723841, -902025631, 15198774361, -258241785517, 4617616213393, -92675891536411, 2008139896617121, -44077045165245841, 1009237851013130497, -25134834585692446903, 658430145120963410281

Offset: 0

Seiichi Manyama, Oct 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..440

Column k=3 of A294289.

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

A294292 Expansion of e.g.f. exp(1/((1+x)(1+x^2)(1+x^3)*(1+x^4)) - 1).

Original entry on oeis.org

1, -1, 1, -7, 49, -181, 1561, -18691, 173377, -2150569, 28355761, -356432671, 5780298481, -95621235997, 1506345546889, -29605056916891, 624987953076481, -12495650070557521, 277892162543608417, -6701525504691986359, 160540685175500028721

Offset: 0

Seiichi Manyama, Oct 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..442

Column k=4 of A294289.

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

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

A294290 E.g.f.: exp(1/((1+x)*(1+x^2)) - 1).

Original entry on oeis.org

1, -1, 1, -1, 25, -241, 1081, -3361, 68881, -1288225, 11828881, -69917761, 1347298921, -36402297361, 533785676425, -4949573821921, 96936811739041, -3274354780495681, 67608887254849441, -885780921100074625, 18450482854023522361, -718444927697360335921

Offset: 0

Seiichi Manyama, Oct 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..446

Column k=2 of A294289.

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

A294293 E.g.f.: exp(1/((1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)) - 1).

Original entry on oeis.org

1, -1, 1, -7, 49, -301, 3001, -26251, 280897, -4040569, 52124401, -749335951, 12646748401, -215856019237, 4053285942889, -82462867482451, 1728838444704001, -38850070313151601, 922947645744036577, -22864122050534606359, 599150979583749111601

Offset: 0

Seiichi Manyama, Oct 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..438

Column k=5 of A294289.

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

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

Views

Author

Keywords

Links

Crossrefs

Formula

A294291 E.g.f.: exp(1/((1+x)*(1+x^2)*(1+x^3)) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A294292 Expansion of e.g.f. exp(1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A294290 E.g.f.: exp(1/((1+x)*(1+x^2)) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A294293 E.g.f.: exp(1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A294291 E.g.f.: exp(1/((1+x)(1+x^2)(1+x^3)) - 1).

A294292 Expansion of e.g.f. exp(1/((1+x)(1+x^2)(1+x^3)*(1+x^4)) - 1).

A294293 E.g.f.: exp(1/((1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)) - 1).