A294212 -id:A294212 - cp's OEIS frontend

A294289 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/(1+x^j) - 1).

1, 1, 0, 1, -1, 0, 1, -1, 3, 0, 1, -1, 1, -13, 0, 1, -1, 1, -1, 73, 0, 1, -1, 1, -7, 25, -501, 0, 1, -1, 1, -7, 73, -241, 4051, 0, 1, -1, 1, -7, 49, -421, 1081, -37633, 0, 1, -1, 1, -7, 49, -181, 2641, -3361, 394353, 0, 1, -1, 1, -7, 49, -301, 1561, -32131, 68881

Offset: 0

Seiichi Manyama, Oct 27 2017

			Square array A(n,k) begins:
   1,    1,    1,    1,    1, ...
   0,   -1,   -1,   -1,   -1, ...
   0,    3,    1,    1,    1, ...
   0,  -13,   -1,   -7,   -7, ...
   0,   73,   25,   73,   49, ...
   0, -501, -241, -421, -181, ...

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

Columns k=0..5 give A000007, A293125, A294290, A294291, A294292, A294293.
Rows n=0 gives A000012.
Main diagonal gives A294261.
Cf. A294212.

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.

A294214 E.g.f.: exp(1/((1-x)(1-x^2)(1-x^3)) - 1).

Original entry on oeis.org

1, 1, 5, 31, 241, 2261, 25501, 319915, 4564001, 71905321, 1240694101, 23250921431, 470598127825, 10200501671101, 236040247113581, 5800885227542371, 150850086300786241, 4137020164029442385, 119309846230265324581, 3608164806033723494671

Offset: 0

Seiichi Manyama, Oct 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..430

Column k=3 of A294212.

nmax = 20; CoefficientList[Series[E^(1/((1-x)*(1-x^2)*(1-x^3)) - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Dec 02 2021 *)

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

a(n) = a(n-1) + 2*(n-1)*n*a(n-2) + (n-2)*(n-1)*(2*n+1)*a(n-3) - (n-10)*(n-3)*(n-2)*(n-1)*a(n-4) - 4*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6) + 2*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7) + 2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-8) - (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-10). - Vaclav Kotesovec, Dec 02 2021

a(n) ~ exp(-101/144 + 29*n^(1/4)/(36*2^(3/4)) + sqrt(n/2) + 2^(7/4)*n^(3/4)/3 - n) * n^(n - 1/8) / 2^(9/8) * (1 + 71323/(103680*2^(1/4)*n^(1/4))). - Vaclav Kotesovec, Dec 02 2021

A294215 E.g.f.: exp(1/((1-x)(1-x^2)(1-x^3)*(1-x^4)) - 1).

Original entry on oeis.org

1, 1, 5, 31, 265, 2501, 29461, 383755, 5721521, 93393865, 1683745381, 32835673751, 693498302905, 15671281854541, 378500195728565, 9704429057721091, 263513260349418721, 7544370749942882705, 227236831102901587141, 7177550671651275241615

Offset: 0

Seiichi Manyama, Oct 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..427

Column k=4 of A294212.

With[{nn=20},CoefficientList[Series[Exp[1/((1-x)(1-x^2)(1-x^3)(1-x^4))-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 08 2019 *)

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

a(n+16) = (n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*n*a(n) - (n+15)*(n+14)*(n+13)*(n+12)*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)*a(n+2) - 2*(n+15)*(n+14)*(n+13)*(n+12)*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*a(n+3) - 2*(n+15)*(n+14)*(n+13)*(n+12)*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*a(n+4) + 2*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+15)*(n+14)*(n+13)*(n+12)*a(n+5) + 3*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+6)*(n+15)*(n+14)*(n+13)*(n+12)*a(n+6) + 4*(n+11)*(n+10)*(n+9)*(n+8)*(n+7)*(n+15)*(n+14)*(n+13)*(n+12)*a(n+7) - 4*(n+11)*(n+10)*(n+9)*(n+15)*(n+14)*(n+13)*(n+12)*a(n+9) - (n+15)*(n+14)*(n+13)*(n+12)*(n+11)*(3*n+20)*a(n+10) - (2*n+11)*(n+15)*(n+14)*(n+13)*(n+12)*a(n+11) + 2*(n+19)*(n+15)*(n+14)*(n+13)*a(n+12) + 2*(n+15)*(n+14)*(n+17)*a(n+13) + (n+18)*(n+15)*a(n+14) + a(n+15). - Robert Israel, Mar 12 2020

a(n) ~ exp(-68413/92160 + 295*n^(1/5) / (192*6^(4/5)) + 15*3^(2/5)*n^(2/5) / (32*2^(3/5)) + 5*n^(3/5) / (4*6^(2/5)) + 5*n^(4/5) / (4*6^(1/5)) - n) * n^(n - 1/10) / (sqrt(5) * 6^(1/10)) * (1 + 18025/(18432*6^(1/5)*n^(1/5))). - Vaclav Kotesovec, Dec 02 2021

A294216 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, 5, 31, 265, 2621, 30901, 411475, 6232241, 103992985, 1909517221, 38038551431, 819452533945, 18924054045781, 466804248205205, 12234394315501051, 339537099805667041, 9941204417393228465, 306212116909615474501, 9894769026271957204975

Offset: 0

Seiichi Manyama, Oct 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..425
Vaclav Kotesovec, Recurrence (of order 25)

Column k=5 of A294212.

With[{nn=20},CoefficientList[Series[Exp[1/Times@@(1-x^Range[5])-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 08 2018 *)

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)))

A294213 E.g.f.: exp(1/((1-x)*(1-x^2)) - 1).

Original entry on oeis.org

1, 1, 5, 25, 193, 1601, 16741, 190345, 2509025, 35825473, 569012581, 9716400761, 180303804385, 3569527588225, 75681964322693, 1700163418683241, 40499757023856961, 1016190431274596225, 26843084299482509125, 743180975111364212953

Offset: 0

Seiichi Manyama, Oct 25 2017

Seiichi Manyama, Table of n, a(n) for n = 0..435

Column k=2 of A294212.

nmax = 20; CoefficientList[Series[E^(1/((1-x)*(1-x^2)) - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 26 2017 *)

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

a(n) ~ exp(-61/96 + 3*n^(1/3)/4 + 3*n^(2/3)/2 - n) * n^(n - 1/6) / sqrt(3) * (1 + 25/(64*n^(1/3))). - Vaclav Kotesovec, Oct 26 2017
a(n) = n*a(n-1) + (n-1)*(2*n - 1)*a(n-2) - 2*(n-3)*(n-2)*(n-1)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - Vaclav Kotesovec, Dec 02 2021
From Peter Bala, Oct 17 2023: (Start)
a(n+k) == a(n) (mod k) for all n and k >= 1. Hence for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k. Cf. A047974.
a(5*n + 2) == a(5*n + 3) == 0 (mod 5);
a(25*n + 3) == a(25*n + 8) == a(25*n + 13) == a(25*n + 17) == a(25*n + 18) == a(25*n + 23) == 0 (mod 5^2);
a(125*n + 18) == a(125*n + 67) == a(125*n + 93) == a(125*n + 118) == 0 (mod 5^3). (End)

cp's OEIS Frontend

A294289 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/(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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A294214 E.g.f.: exp(1/((1-x)*(1-x^2)*(1-x^3)) - 1).

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A294215 E.g.f.: exp(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) - 1).

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A294216 E.g.f.: exp(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) - 1).

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A294213 E.g.f.: exp(1/((1-x)*(1-x^2)) - 1).

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A294214 E.g.f.: exp(1/((1-x)(1-x^2)(1-x^3)) - 1).

A294215 E.g.f.: exp(1/((1-x)(1-x^2)(1-x^3)*(1-x^4)) - 1).

A294216 E.g.f.: exp(1/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)) - 1).