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 1-4 of 4 results.

A337797 Number of partitions of the n-th n-gonal pyramidal number into n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 2, 4, 13, 45, 198, 858, 3728, 16115, 69125, 292940, 1224628, 5052396, 20570806, 82655098, 327881398, 1284663878, 4973614490, 19034194696, 72037124003, 269723590850, 999517370314, 3667158097572, 13325691939021, 47975192145998
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

Examples

			a(3) = 4 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A006484, A072964, A298269, A337762, A337798, A337799.

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (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.

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

A336091 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n n-gonal pyramidal numbers (with 0's allowed).

Original entry on oeis.org

1, 1, 2, 3, 10, 5, 246, 1519, 19678, 74601, 690490, 21026621, 301528272, 4397123315, 71221546592, 1001245733295, 19276579678736, 368677642975493, 6820451221691646, 136000924000323691, 3069656935024721420, 69646109074231173897, 1641880679174919030100
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 04 2020

Keywords

Examples

			a(3) = 3 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10, 0, 0], [0, 10, 0] and [0, 0, 10].

Links

Crossrefs

Cf. A006484, A196010, A335633, A336303, A337797, A337798, A337799.

Programs

  • Mathematica
    Table[SeriesCoefficient[Sum[x^(k (k + 1) (k (n - 2) - n + 5)/6), {k, 0, n}]^n, {x, 0, n (n + 1) (n^2 - 3 n + 5)/6}], {n, 0, 22}]

Formula

a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

A336303 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n nonzero n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 0, 0, 6, 0, 180, 630, 1120, 36288, 441000, 6579870, 59734620, 1252872192, 13668490836, 162131872695, 2971275208720, 52783774330940, 1334562954639156, 16933262255752698, 406499325562503480, 8838644883526856832, 190698441426122689290
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 04 2020

Keywords

Examples

			a(4) = 6 because the fourth square pyramidal number is 30 and we have [14, 14, 1, 1], [14, 1, 14, 1], [14, 1, 1, 14], [1, 14, 14, 1], [1, 14, 1, 14] and [1, 1, 14, 14].

Links

Crossrefs

Cf. A006484, A298858, A335634, A336091, A337797, A337798, A337799.

Programs

  • Mathematica
    Join[{1},Table[SeriesCoefficient[Sum[x^(k (k + 1) (k (n - 2) - n + 5)/6), {k, 1, n}]^n, {x, 0, n (n + 1) (n^2 - 3 n + 5)/6}], {n, 1, 22}]]

Formula

a(n) = [x^p(n,n)] (Sum_{k=1..n} x^p(n,k))^n, where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.
Showing 1-4 of 4 results.

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