A028377 -id:A028377 - cp's OEIS frontend

A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

1, 3, 9, 29, 78, 207, 526, 1284, 3054, 7084, 16071, 35748, 78167, 168195, 356754, 746772, 1544145, 3157056, 6387114, 12795366, 25397760, 49977262, 97542936, 188912466, 363196750, 693424803, 1315161528, 2478648920, 4643337213, 8648452782, 16019345259, 29515269060, 54104712129

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000217, A027480, A028377, A217093, A219555, A343200, A344097, A344098.

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 2, 2], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 2, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 32}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A027480(d) ) * a(n-k).

a(n) ~ (7/15)^(1/8) * 2^(-21/8) * n^(-5/8) * exp((2/3)*(7/15)^(1/4)*Pi * n^(3/4) + 9*sqrt(15/7)*zeta(3) * sqrt(n) / (2*Pi^2) + ((5/7)^(1/4)*Pi / (2*3^(3/4)) - 1215*(15/7)^(1/4)*zeta(3)^2 / (28*Pi^5)) * n^(1/4) + 54675*zeta(3)^3 / (98*Pi^8) - 45*zeta(3) / (28*Pi^2)). - Vaclav Kotesovec, May 12 2021

A294839 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-1)/2)(1 + x^(2k))^(k(3*k+1)/2).

Original entry on oeis.org

1, 1, 2, 7, 13, 30, 61, 125, 250, 494, 960, 1835, 3487, 6520, 12105, 22239, 40515, 73207, 131315, 233831, 413625, 727100, 1270405, 2207243, 3814155, 6557164, 11217391, 19099932, 32375026, 54640509, 91836697, 153739008, 256379360, 425964293, 705197513, 1163452547, 1913096832, 3135609791

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized pentagonal numbers (A001318).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A001318, A028377, A294102, A294840, A294841.

nmax = 37; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 1)/2) (1 + x^(2 k))^(k (3 k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 37}]

G.f.: Product_{k>=1} (1 + x^k)^A001318(k).

a(n) ~ exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / (3*5^(1/4)) + 9*Zeta(3) * sqrt(5*n/7) / (4*Pi^2) + (7*Pi^6 - 2430*Zeta(3)^2) * (5/7)^(1/4) * n^(1/4) / (336 * sqrt(2) * Pi^5) + 15*Zeta(3)*(3240*Zeta(3)^2 - 7*Pi^6) / (3136*Pi^8)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

A294840 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(5k-3)/2)(1 + x^(2k))^(k(5*k+3)/2).

Original entry on oeis.org

1, 1, 4, 11, 26, 65, 150, 343, 760, 1670, 3574, 7561, 15752, 32396, 65850, 132386, 263447, 519316, 1014744, 1966234, 3780464, 7215020, 13674227, 25744768, 48166429, 89576421, 165638008, 304615115, 557275053, 1014398476, 1837617957, 3313527482, 5948262037, 10632231253, 18926026208

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized heptagonal numbers (A085787).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A028377, A085787, A294837, A294839, A294841.

nmax = 34; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (5 k - 3)/2) (1 + x^(2 k))^(k (5 k + 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} (1 + x^k)^A085787(k).

a(n) ~ 7^(1/8) * exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / 3^(5/4) + 15*Zeta(3) * sqrt(3*n/7) / (4*Pi^2) - (7*Pi^6 + 4050*Zeta(3)^2)*n^(1/4) / (112*sqrt(2) * 3^(3/4) * 7^(1/4) * Pi^5) + 15*Zeta(3) * (7*Pi^6 + 5400*Zeta(3)^2) / (3136*Pi^8)) / (2^(7/3) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).

Original entry on oeis.org

1, 1, 5, 13, 34, 87, 212, 504, 1167, 2665, 5933, 13042, 28191, 60148, 126688, 263821, 543414, 1108272, 2239182, 4484482, 8907530, 17555485, 34345465, 66724969, 128772908, 246951514, 470738283, 892159198, 1681544803, 3152656375, 5880839454, 10916463171, 20169007200, 37095527149

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized octagonal numbers (A001082).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A001082, A028377, A294838, A294839, A294840.

nmax = 33; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 2)) (1 + x^(2 k))^(k (3 k + 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A001082(k+1).

a(n) ~ exp(Pi/3 * (7/5)^(1/4) * 2^(3/4) * n^(3/4) + 9*Zeta(3) / (2*Pi^2) * sqrt(5*n/14) - (405*Zeta(3)^2 / (56*Pi^5) + Pi/48) * (10*n/7)^(1/4) + (6075*Zeta(3)^2 / (196*Pi^8) + 15/(224*Pi^2)) * Zeta(3)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 2, 9, 31, 127, 494, 1994, 8040, 32741, 133855, 549775, 2266756, 9372300, 38862245, 161500403, 672538548, 2805669061, 11723319333, 49055511943, 205534846202, 862167483656, 3620429584614, 15217780335870, 64022149180478, 269566679312520, 1135878674712355

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the weigh transform of n-gonal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A027998, A028377, A294102, A294836, A294837, A294838, A318118, A318125.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.2950655312028649462400... and c = 0.204576644650802181512... - Vaclav Kotesovec, Aug 19 2018

A028376 Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 2, 3, 5, 0, 5, 7, 9, 14, 0, 14, 19, 23, 28, 42, 0, 42, 56, 66, 76, 90, 132, 0, 132, 174, 202, 227, 255, 297, 429, 0, 429, 561, 645, 715, 785, 869, 1001, 1430, 0, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 0, 4862, 6292, 7150

Offset: 0

Wouter Meeussen

			0;
0,1;
0,1,2;
0,2,3,5;
0,5,7,9,14;
0,14,19,23,28,42; ...

See A028364 for a better version. Cf. A028377, A028378.

A295180 Expansion of Product_{k>=1} (1 + x^k)^(3k(k-1)/2+1).

Original entry on oeis.org

1, 1, 4, 14, 35, 96, 242, 609, 1483, 3565, 8376, 19389, 44254, 99584, 221470, 486810, 1058914, 2280519, 4866492, 10294313, 21598679, 44966391, 92930485, 190721585, 388828094, 787710401, 1586166758, 3175548134, 6322372729, 12520759979, 24669499432, 48367447687, 94381633962, 183331308393

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

Weigh transform of the centered triangular numbers (A005448).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -(3*n*(n-1)/2+1), g(n) = -1. - Seiichi Manyama, Nov 16 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A028377, A258343, A295179.

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(3 k (k - 1)/2 + 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A005448(k).
a(n) ~ exp(15*Zeta(3) / (28*Pi^2) - 6075*Zeta(3)^3 / (98*Pi^8) + (Pi/6 - 405*Zeta(3)^2 / (28*Pi^5)) * (5*n/7)^(1/4) - (9*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2*Pi * (7/5)^(1/4)/3) * n^(3/4)) * 7^(1/8) / (2^(19/8) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 16 2017
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(3*d*(d-1)/2+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 16 2017

A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

Original entry on oeis.org

1, 2, 8, 26, 76, 216, 590, 1554, 3988, 9988, 24464, 58794, 138866, 322808, 739658, 1672372, 3734848, 8245956, 18012114, 38952586, 83448832, 177194716, 373111970, 779430870, 1615995262, 3326484686, 6800794428, 13813260736, 27881653590, 55942340000, 111601021856

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A000294 and A028377.

N. J. A. Sloane, Transforms

Cf. A000217, A000294, A015128, A028377, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000217(k).
a(n) ~ exp(2*Pi*n^(3/4)/3 + 7*Zeta(3)*sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) + 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3)/(8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(25/12) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^3) ). Cf. A000122 and A156616. - Peter Bala, Dec 23 2021

A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

Original entry on oeis.org

1, 1, 5, 20, 60, 190, 561, 1651, 4720, 13300, 36716, 99872, 267836, 708890, 1854255, 4796273, 12279445, 31135188, 78236006, 194921680, 481758832, 1181675902, 2877646681, 6959866116, 16723591530, 39934902812, 94795718409, 223741936855, 525206126933, 1226393510220

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000332, A000391, A028377, A258343, A305206, A344100, A344101.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^5) ).

A344100 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).

Original entry on oeis.org

1, 1, 6, 27, 92, 323, 1070, 3527, 11314, 35708, 110478, 336629, 1011097, 2997233, 8778761, 25424358, 72867447, 206804742, 581573340, 1621407554, 4483701126, 12303384015, 33514076529, 90656680725, 243603875523, 650444927010, 1726229294595, 4554686670838, 11950683658941

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000389, A000417, A028377, A258343, A305206, A344099, A344101.

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 5], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 4, 5], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^6) ).

cp's OEIS Frontend

A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

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A294839 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-1)/2)*(1 + x^(2*k))^(k*(3*k+1)/2).

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A294840 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(5*k-3)/2)*(1 + x^(2*k))^(k*(5*k+3)/2).

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A294841 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-2))*(1 + x^(2*k))^(k*(3*k+2)).

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A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

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A028376 Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

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A295180 Expansion of Product_{k>=1} (1 + x^k)^(3*k*(k-1)/2+1).

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A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

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A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

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A294839 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-1)/2)(1 + x^(2k))^(k(3*k+1)/2).

A294840 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(5k-3)/2)(1 + x^(2k))^(k(5*k+3)/2).

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).

A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^3)).

A295180 Expansion of Product_{k>=1} (1 + x^k)^(3k(k-1)/2+1).