A003406 -id:A003406 - cp's OEIS frontend

A306704 Expansion of Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 + jx^j).

1, 1, -1, 2, -2, 0, 1, 3, -5, -6, 11, 11, -12, -35, 33, 35, -22, -102, 170, 47, -224, -491, 874, 695, -598, -2606, 2246, 1503, -664, -6420, 11590, 2526, -13762, -34647, 61785, 37119, -32372, -181052, 147105, 104896, 12824, -436333, 799007, -109587, -868230, -2316921, 4447531

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A003406, A204856.

nmax = 46; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306706 Expansion of Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, -1, 2, -2, 0, 1, 2, -4, -3, 7, 4, -4, -14, 6, 10, 12, -14, 0, -26, 3, 7, 60, 11, -27, -99, -26, 6, 126, 94, 58, -87, -180, -201, -46, 145, 282, 330, 142, -21, -515, -573, -716, -15, 423, 1519, 1128, 1197, -783, -1378, -3264, -1892, -1574, 2155, 2679, 6075, 3376, 3243

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A003406, A206138.

nmax = 57; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A302580 Numbers k such that the coefficient of x^k in the expansion of Ramanujan's function R(x) = Sum_{i>=0} x^(i*(i+1)/2)/Product_{j=1..i} (1 + x^j) is zero.

Original entry on oeis.org

6, 9, 11, 16, 20, 21, 23, 27, 29, 30, 31, 33, 34, 36, 37, 38, 41, 44, 46, 49, 53, 56, 58, 59, 60, 61, 63, 64, 65, 66, 71, 72, 79, 80, 81, 82, 85, 86, 91, 93, 94, 96, 97, 98, 102, 104, 106, 107, 110, 111, 114, 115, 116, 120, 121, 122, 124, 128, 129, 131, 133, 135, 136, 137, 141, 142, 146, 148

Offset: 1

Ilya Gutkovskiy, Apr 10 2018

nonn

Numbers k such that number of partitions of k into distinct parts with even rank equals number of partitions of k into distinct parts with odd rank (the rank of a partition is its largest part minus the number of parts).

Index entries for sequences related to partitions

Cf. A003406, A117192, A117193.

Flatten[Position[nmax = 150; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2)/Product[(1 + x^j), {j, 1, i}], {i, 0, nmax}], {x, 0, nmax}], x]], 0]]

cp's OEIS Frontend

A306704 Expansion of Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 + jx^j).

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A306706 Expansion of Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 + x^j)^j.

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A302580 Numbers k such that the coefficient of x^k in the expansion of Ramanujan's function R(x) = Sum_{i>=0} x^(i*(i+1)/2)/Product_{j=1..i} (1 + x^j) is zero.

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cp's OEIS Frontend

A306704 Expansion of Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 + j*x^j).

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A306706 Expansion of Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 + x^j)^j.

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A302580 Numbers k such that the coefficient of x^k in the expansion of Ramanujan's function R(x) = Sum_{i>=0} x^(i*(i+1)/2)/Product_{j=1..i} (1 + x^j) is zero.

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A306704 Expansion of Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 + jx^j).