A003633 -id:A003633 - cp's OEIS frontend

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 10, 11, 11, 12, 12, 14, 15, 15, 16, 16, 18, 19, 19, 21, 22, 24, 26, 26, 28, 29, 31, 33, 33, 35, 36, 39, 42, 43, 45, 47, 50, 53, 54, 56, 58, 61, 65, 66, 69, 72, 76, 81, 83, 86, 89, 93, 98, 100, 103, 107, 112, 118, 121, 125, 130, 136, 142, 146

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered square numbers (A001844).

			a(10) = 3 because we have [5, 5], [5, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

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]
Eric Weisstein's World of Mathematics, Centered Square Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A001156, A001844, A279220, A280950, A280952, A280953.

nmax = 82; CoefficientList[Series[Product[1/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

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

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 67, 68, 71, 74, 77, 79, 83, 85, 88, 91, 94, 96, 100

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered pentagonal numbers (A005891).

			a(12) = 3 because we have [6, 6], [6, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
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]
Eric Weisstein's World of Mathematics, Centered Pentagonal Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005891, A218379, A279221, A280950, A280951, A280953.

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(((t+1)*5*t+2)/2>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(((i+1)*5*i+2)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax = 88; CoefficientList[Series[Product[1/(1 - x^(5 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 22, 23, 24, 25, 25, 27, 27, 29, 30, 31, 32, 32, 34, 34, 36, 37, 38, 39, 40, 43, 44, 46, 47, 48, 50, 51, 54, 55, 57, 58, 59

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered hexagonal numbers (A003215).

			a(14) = 3 because we have [7, 7], [7, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
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]
Eric Weisstein's World of Mathematics, Hex Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A003215, A278949, A279222, A280950, A280951, A280952.

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(3*t*(t+1)+1>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(3*i*(i+1)+1)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax = 86; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

A281156 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k+1)/6).

Original entry on oeis.org

1, 1, 5, 19, 54, 165, 467, 1317, 3599, 9687, 25519, 66203, 169254, 426750, 1062950, 2616818, 6373911, 15369774, 36716706, 86939235, 204152395, 475631501, 1099874363, 2525418842, 5759549109, 13050991205, 29391523405, 65801951770, 146486952644, 324340095729, 714389015139

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Weigh transform of square pyramidal numbers (A000330).

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, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A027998, A258343.

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).

a(n) ~ exp(5*(15*Zeta(5))^(1/5) * n^(4/5) / 2^(11/5) + 7*Pi^4 * n^(3/5) / (360*2^(2/5) * (15*Zeta(5))^(3/5)) + (Zeta(3) / (2^(13/5) * (15*Zeta(5))^(2/5)) - 49*Pi^8 / (2160000 * 2^(3/5) * 15^(2/5) * Zeta(5)^(7/5)))*n^(2/5) + (343*Pi^12 / (9720000000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + 49*Pi^8 * Zeta(3) / (129600000 * Zeta(5)^2) - 2401 * Pi^16 / (83980800000000 * Zeta(5)^3) - Zeta(3)^2 / (1200*Zeta(5))) * (3*Zeta(5))^(1/10) / (2^(11/18) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017

A294836 Expansion of Product_{k>=1} (1 + x^k)^(k(2k-1)).

Original entry on oeis.org

1, 1, 6, 21, 58, 178, 494, 1365, 3640, 9533, 24401, 61384, 151958, 370335, 890565, 2113913, 4959199, 11505799, 26420628, 60082005, 135386341, 302448477, 670148898, 1473387787, 3215519032, 6968266907, 14999453058, 32079714584, 68187859040, 144083404856, 302727633735, 632579826174

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the hexagonal numbers (A000384).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(2*n-1), g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..9392
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, Hexagonal Number

Cf. A000384, A027998, A028377, A294102, A294837, A294838.

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: Product_{k>=1} (1 + x^k)^A000384(k).
a(n) ~ 7^(1/8) * exp(Pi*2^(3/2) * (7/15)^(1/4) * n^(3/4)/3 - 3*Zeta(3)*sqrt(15*n/7) / (2*Pi^2) - 135*Zeta(3)^2 * (15*n/7)^(1/4) / (28*sqrt(2)*Pi^5) - 2025*Zeta(3)^3 / (196*Pi^8)) / (2^(5/3) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 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^2*(2*d-1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A300452 Logarithmic transform of the cubes A000578.

Original entry on oeis.org

0, 1, 7, 5, -146, -351, 9936, 51421, -1394000, -12844287, 328407400, 4874111901, -115361217696, -2607873466511, 55768370301112, 1866984952934445, -34886452149332864, -1720211491314549375, 26716801597874981064, 1979492625918149729437, -23490293022369696366560, -2777285149336544358953679

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			E.g.f.: A(x) = x/1! + 7*x^2/2! + 5*x^3/3! - 146*x^4/4! - 351*x^5/5! + 9936*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..408
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, Logarithmic Transform
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A009306, A033464, A279358.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i^3)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

nmax = 21; CoefficientList[Series[Log[1 + Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: log(1 + exp(x)*x*(1 + 3*x + x^2)).

A007469 Shifts left 2 places under Stirling2 transform.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 66, 343, 2167, 16193, 140919, 1414947, 16258868, 211935996, 3105828560, 50748310068, 918138961643, 18287966027343, 399145502051200, 9505803743367971, 246064556796896554, 6897674469134480653, 208651954748397405264, 6788671409470892058148

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..150
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

stirtr:= proc(p)
           proc(n) add(p(k)*Stirling2(n, k), k=0..n) end
         end:
a:= proc(n) option remember; `if`(n<3, 1, aa(n-2)) end:
aa:= stirtr(a):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 22 2012

stirtr[p_] := Function[{n}, Sum[p[k]*StirlingS2[n, k], {k, 0, n}]]; a[n_] := a[n] = If[n<3, 1, aa[n-2]]; aa = stirtr[a]; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Jan 09 2013, translated from Alois P. Heinz's Maple program *)

A007553 Logarithmic transform of Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 7, 5, 85, 335, 1135, 15245, 13475, 717575, 4256825, 29782325, 525045275, 243258625, 56809006625, 415670267875, 5068080417875, 104229929847625, 60861649495625, 20784245979986875, 169274937975443125, 3318579283890780625, 75028912866554839375

Offset: 1

N. J. A. Sloane

nonn

The coefficients of the e.g.f. log(Sum_{n>=0} Fibonacci(n+1)*x^n/n!) produce the sequence [1,1,-1,-1,7,-5,-85,...], offset 0. - Peter Bala, Jan 19 2011
The series reversion of Sum_{n>=1} Fibonacci(n)*x^n/n is an e.g.f. whose coefficient sequence [1,-1,-1,7,-5,-85,335,1135,...] (offset 1) appears to be a signed version of this sequence. - Peter Bala, Jan 19 2011
E.g.f. A(x), A(x)=x*B(x) satisfies the differential equation B'(x) = 1 + B(x) - B(x)*B(x). - Vladimir Kruchinin, Nov 03 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..340
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
Index entries for sequences related to logarithmic numbers

b:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*t(n-j)*
      binomial(n, j)*b(j), j=1..n-1)/n))(i->(<<0|1>, <1|1>>^i)[2, 2])
    end:
a:= n-> abs(b(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 06 2018

FullSimplify[Abs[Rest[CoefficientList[Series[-2*x/(1+Sqrt[5]) - Log[5+Sqrt[5]] + Log[2+(3+Sqrt[5])*E^(Sqrt[5]*x)], {x, 0, 15}], x] * Range[0, 15]!]]] (* Vaclav Kotesovec, Jun 24 2014 *)

b(n):=if n=0 then 1 else b(n-1)-sum(b(i)*b(n-1-i)*binomial(n-1,i),i,1,n-2);
a(n):=if n=0 then 0 else abs(b(n-1)); # Vladimir Kruchinin, Nov 03 2011

b(n):=if n=1 then 1 else sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum(((sqrt(5)+1)^(n+j-i-1)*5^((i-j)/2)*stirling1(i,j)*2^(-n-j+i+1)*binomial(n+j-2,i-1))/i!,i,j,n+j-1)),j,1,k),k,1,n-1);
a(n):=if n=1 then 1 else abs(b(n-1));
makelist(ratsimp(a(n)),n,1,10); # Vladimir Kruchinin, Nov 17 2012

@CachedFunction
def c(n,k) :
    if n==k: return 1
    if k<1 or k>n: return 0
    return ((n-k)//2+1)*c(n-1,k-1)-2*k*c(n-1,k+1)
@CachedFunction
def A007553(n):
    return abs(add(c(n,k) for k in (0..n)))
[A007553(n) for n in (0..25)] # Peter Luschny, Jun 10 2014

b(n) = b(n-1) - Sum_{i=1..n-2} b(i)*b(n-1-i)*binomial(n-1,i), b(0)=1. a(n+1) = abs(b(n)). - Vladimir Kruchinin, Nov 03 2011
Let e.g.f. E(x) = log(1 + Sum_{n>=1} Fibonacci(n+1)*x^n/n!), then g.f. A(x)=x*(1+1/Q(0)), where Q(k) = 1/(x*(k+1)) + 1 + 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
Let F(x) = log(Sum_{n>=0} Fibonacci(n+1)*x^n/n!) be the e.g.f., produce the sequence [1,1,-1,-1,7,-5,-85,...], then g.f. A(x)= 1 + x/Q(0), where Q(k) = 1 + x*(2*k+1) + x^2*(2*k+1)*(2*k+2)/(1 + x*(2*k+2) + x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013
a(n) ~ 2*(n-1)! * abs(cos(n*arctan(Pi/log(2/(3+sqrt(5)))))) * (5/(Pi^2+log(2/(3+sqrt(5)))^2))^(n/2). - Vaclav Kotesovec, Jun 24 2014

A112005 Logarithmic transform of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 0, 1, -2, 4, -17, 82, -384, 2189, -14850, 107404, -845537, 7400482, -70093256, 709888645, -7721333538, 89774204756, -1107347563761, 14456268008050, -199350032354000, 2893615098314941, -44089764970860290, 703841452185590236, -11747695951762870497

Offset: 0

Wolfdieter Lang, Sep 12 2005

Alois P. Heinz, Table of n, a(n) for n = 0..200
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
Index entries for sequences related to logarithmic numbers

Cf. A007553, A112006, A256180.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*t(n-j)*
      binomial(n, j)*a(j), j=1..n-1)/n))(i->(<<0|1>, <1|1>>^i)[1, 2])
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

FullSimplify[CoefficientList[Series[Log[1 + 2*E^(x/2)*Sinh[Sqrt[5]*x/2] / Sqrt[5]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Sep 04 2014 *)

E.g.f. log(1 + A(x)) with the e.g.f. A(x):=exp(x/2)*sinh(sqrt(5)*x/2)/(sqrt(5)/2) of A000045.

a(n) ~ -(n-1)! / r^n, where r = -1.37807491378452630283968362340785266756... is the root of the equation 2*(5-3*sqrt(5))*r + (sqrt(5)-5) * (log(5/4) + 2*log(1-coth(sqrt(5)*r/2))) = 0. - Vaclav Kotesovec, Sep 04 2014

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 24, 25, 26, 26, 27, 28, 29, 31, 32, 33, 33, 34, 35, 37, 39, 41, 42, 43, 45, 46, 48, 50, 52, 53, 54, 56, 58, 60, 62, 64, 65, 67, 69, 72, 75, 78

Offset: 0

Ilya Gutkovskiy, Dec 03 2016

Number of partitions of n into nonzero heptagonal numbers (A000566).

			a(8) = 2 because we have [7, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
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]
Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000566, A001156, A007294, A037444, A218379, A278949.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(5*t-3)/2>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(5*i-3)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (5 k - 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(5*k-3)/2)).

cp's OEIS Frontend

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2*k*(k+1)+1)).

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A280952 Expansion of Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

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A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

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A281156 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).

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A294836 Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k-1)).

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A300452 Logarithmic transform of the cubes A000578.

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Maple

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A007469 Shifts left 2 places under Stirling2 transform.

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Maple

Mathematica

A007553 Logarithmic transform of Fibonacci numbers.

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A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

A281156 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k+1)/6).

A294836 Expansion of Product_{k>=1} (1 + x^k)^(k(2k-1)).

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).