A206100 -id:A206100 - cp's OEIS frontend

A206138 G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k)^k.

1, 1, 1, 2, 2, 4, 5, 8, 10, 17, 21, 32, 44, 62, 86, 122, 164, 230, 318, 428, 591, 803, 1088, 1467, 1995, 2665, 3596, 4800, 6430, 8552, 11416, 15093, 20062, 26487, 34988, 46035, 60626, 79490, 104278, 136337, 178189, 232331, 302724, 393493, 511165, 662775, 858380

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 ...
where
A(x) = 1 + x/(1-x) + x^3/((1-x)*(1-x^2)^2) + x^6/((1-x)*(1-x^2)^2*(1-x^3)^3) + x^10/((1-x)*(1-x^2)^2*(1-x^3)^3*(1-x^4)^4) +...

Cf. A206100.

{a(n)=polcoeff(sum(m=0,n,x^(m*(m+1)/2)/prod(k=1,m,(1-x^k +x*O(x^n))^k)),n)}
for(n=0,60,print1(a(n),", "))

A318771 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^j)^j.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 22, 25, 37, 47, 64, 81, 113, 140, 191, 243, 319, 408, 540, 677, 889, 1132, 1462, 1855, 2404, 3034, 3909, 4946, 6325, 7997, 10202, 12840, 16328, 20549, 25989, 32627, 41180, 51577, 64872, 81128, 101729, 127016, 158913, 197981, 247163, 307523, 383019

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

Cf. A193197, A206100, A206138, A318770.

a:=series(add(x^(k^2)/mul((1-x^j)^j,j=1..k),k=0..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

nmax = 52; CoefficientList[Series[Sum[x^k^2/Product[(1 - x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306703 Expansion of Sum_{k>=0} x^k / Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, 0, 1, -2, 1, -2, 1, -4, 7, -5, 7, -7, 16, -13, 16, -33, 38, -39, 31, -75, 79, -92, 118, -139, 201, -230, 269, -264, 494, -476, 523, -780, 886, -1095, 1261, -1533, 1857, -2593, 2197, -3367, 4256, -4240, 5816, -6484, 7985, -9800, 11051, -12561, 17530, -17516

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A081362, A206100.

nmax = 50; CoefficientList[Series[Sum[x^k/Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306664 Expansion of Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)^j.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 22, 31, 37, 54, 63, 89, 111, 146, 184, 247, 301, 398, 501, 642, 804, 1042, 1293, 1663, 2082, 2648, 3321, 4229, 5268, 6691, 8370, 10553, 13168, 16595, 20659, 25929, 32253, 40321, 50092, 62489, 77418, 96340, 119266, 147998, 182927, 226609

Offset: 0

Ilya Gutkovskiy, Mar 04 2019

nonn

Cf. A003106, A206100, A206138.

nmax = 54; CoefficientList[Series[Sum[x^(k (k + 1))/Product[(1 - x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306731 Expansion of Sum_{k>=0} x^k * Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 10, 14, 24, 37, 56, 90, 135, 204, 316, 474, 704, 1068, 1579, 2332, 3445, 5054, 7376, 10750, 15587, 22497, 32437, 46544, 66520, 94908, 134912, 191185, 270301, 380924, 535469, 750898, 1050268, 1465284, 2039741, 2832694, 3925036, 5427381, 7488315, 10310431

Offset: 0

Ilya Gutkovskiy, Mar 06 2019

nonn

Cf. A000009, A206100, A306703.

nmax = 43; CoefficientList[Series[Sum[x^k Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A206138 G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k)^k.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A318771 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^j)^j.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A306703 Expansion of Sum_{k>=0} x^k / Product_{j=1..k} (1 + x^j)^j.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A306664 Expansion of Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)^j.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A306731 Expansion of Sum_{k>=0} x^k * Product_{j=1..k} (1 + x^j)^j.

Views

Author

Keywords

Crossrefs

Programs

Mathematica