cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 21, 56, 144, 370, 926, 2275, 5482, 12966, 30124, 68838, 154934, 343756, 752689, 1627701, 3479226, 7355608, 15390682, 31889732, 65465473, 133212912, 268811363, 538119723, 1069051243, 2108416588, 4129355331, 8033439333
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2020

Keywords

Examples

			a(3) = 4 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000041, A037444, A060354, A072964, A337763, A337764.

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^(k*((k*(n - 2) - n + 4)/2))), {k, 1, n}], {x, 0, n*(4 - 3*n + n^2)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 19 2020 *)

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A337764 Number of compositions (ordered partitions) of the n-th n-gonal number into n-gonal numbers.

Original entry on oeis.org

1, 1, 2, 7, 124, 14371, 12842911, 103590035354, 8621925847489749, 8307493939404888703058, 102488432265617100812550713499, 17706351554929677399562928448484650120, 46435685450659378932235460132506329282776942795
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2020

Keywords

Examples

			a(3) = 7 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A011782, A060354, A224366, A224677, A337762, A337763.

Formula

a(n) = [x^p(n,n)] 1 / (1 - Sum_{k=1..n} x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A337798 Number of partitions of the n-th n-gonal pyramidal number into distinct n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 5, 4, 5, 7, 11, 9, 4, 12, 12, 24, 23, 42, 59, 64, 58, 124, 206, 212, 168, 377, 539, 703, 873, 1122, 1505, 1943, 2724, 4100, 4513, 6090, 7138, 12079, 16584, 20240, 27162, 35874, 52622, 69817, 88059, 115628, 152756, 219538, 240200, 358733, 480674
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

Examples

			a(9) = 2 because the ninth 9-gonal pyramidal number is 885 and we have [885] and [420, 266, 155, 34, 10].

Links

Crossrefs

Cf. A006484, A288126, A298857, A337763, A337797, A337799.

Programs

  • Maple
    p:= (n,k) ->  k * (k + 1) * (k * (n - 2) - n + 5) / 6:
f:= proc(n) local k, P;
  P:= mul(1+x^p(n,k),k=1..n);
  coeff(P,x,p(n,n));
end proc:
map(f, [$0..80]); # Robert Israel, Sep 23 2020
  • PARI
    default(parisizemax, 2^31);
p(n,k) = k*(k + 1)*(k*(n-2) - n + 5)/6;
a(n) = my(f=1+x*O(x^p(n,n))); for(k=1, n, f*=1+x^p(n,k)); polcoeff(f, p(n,n)); \\ Jinyuan Wang, Dec 21 2021

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} (1 + x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

Extensions

More terms from Robert Israel, Sep 23 2020

A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

Original entry on oeis.org

1, 1, 3, 6, 5, 95, 336, 2597, 26832, 197577, 1847800, 14621101, 129754956, 1146534701, 12342194879, 161225146370, 2464561564936, 39642413790129, 620059254486798, 9430493858327959, 136438759335452360, 1881721996407396801, 24999081626667425376, 321601467988647184779
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2020

Keywords

Examples

			a(3) = 6 because the third triangular number is 6 and we have [6, 0, 0], [0, 6, 0], [0, 0, 6], [3, 3, 0], [3, 0, 3] and [0, 3, 3].

Links

Crossrefs

Cf. A060354, A088218, A196010, A298329, A335634, A337762, A337763, A337764.

Programs

  • Mathematica
    Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 0, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 0, 23}]
  • PARI
    p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=0, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

Formula

a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 0, 1, 30, 180, 700, 3780, 11844, 50610, 325820, 5803380, 126594910, 2114901789, 28282722650, 323420067880, 3190581939996, 29336527986960, 245438739897312, 1967485926594030, 16000631392009320, 184418174847183508, 4054670001158799616, 111835386569787369559
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2020

Keywords

Examples

			a(4) = 1 because the fourth square is 16 and we have [4, 4, 4, 4].

Links

Crossrefs

Cf. A060354, A298330, A298858, A335633, A337762, A337763, A337764.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 1, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 1, 24}]]
  • PARI
    p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=1, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

Formula

a(n) = [x^p(n,n)] (Sum_{k=1..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.
Showing 1-5 of 5 results.

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