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 1-3 of 3 results.

A339411 Product of partial sums of odd squares.

Original entry on oeis.org

1, 1, 10, 350, 29400, 4851000, 1387386000, 631260630000, 429257228400000, 415950254319600000, 553213838245068000000, 979741707532015428000000, 2253405927323635484400000000, 6591212337421633791870000000000, 24084289880938649875492980000000000, 108258883014819231190340945100000000000
Offset: 0

Views

Author

Werner Schulte, Dec 03 2020

Keywords

Comments

a(n) is also the number of labeled histories across all trifurcating labeled topologies on trees with non-simultaneous trifurcations, where the number of leaves is 2n+1. - Noah A Rosenberg, Feb 24 2025

Examples

			a(4) = (1^2) * (1^2 + 3^2) * (1^2 + 3^2 + 5^2) * (1^2 + 3^2 + 5^2 + 7^2) = 1 * 10 * 35 * 84 = 29400.

Links

Crossrefs

Cf. A016754, A000447, A135438, A000108, A006472.

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*(4*n^3-n)/3)
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    Array[((2 #)!*(2 # + 1)!)/(#!*12^#) &, 16, 0] (* Michael De Vlieger, Dec 10 2020 *)
  • PARI
    for(n=0,9,print((2*n)!*(2*n+1)!/(n!*12^n)))
  • PARI
    for(n=0,9,print(prod(i=1,n,sum(j=1,i,(2*j-1)^2))))

Formula

a(n) = Product_{i=1..n} (Sum_{j=1..i} (2*j - 1)^2).
a(n) = Product_{i=1..n} binomial(2*i + 1, 3).
a(n) = Product_{i=1..n} A000447(i).
a(n) = ((2*n)! * (2*n+1)!) / (n! * 12^n).
a(n) / A135438(n) = A000108(n).
a(n) = (Gamma(2*n + 2)*Gamma(n + 1/2))/(3^n*sqrt(Pi)). - Peter Luschny, Dec 11 2020
D-finite with recurrence 3*a(n) -n*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 25 2023

A279662 a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

Original entry on oeis.org

1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 16 2016

Keywords

Comments

Hexagonal pyramidal factorial numbers.
More generally, the m-gonal pyramidal factorial numbers is 6^(-n)*(m-2)^n*Gamma(n+1)*Gamma(n+2)*Gamma(n+3/(m-2))/Gamma(3/(m-2)), m>2.

Links

Crossrefs

Cf. A002412.
Cf. A000680 (hexagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279663 (heptagonal pyramidal factorial numbers).

Programs

  • Magma
    [Round((2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2) / Gamma(3/4)): n in [0..20]]; // Vincenzo Librandi, Dec 17 2016
  • Mathematica
    FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]

Formula

a(n) = Product_{k=1..n} k*(k + 1)*(4*k - 1)/6, a(0)=1.
a(n) = Product_{k=1..n} A002412(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(2/3)^n*n^(3*n+9/4)/(Gamma(3/4)*exp(3*n)).

A279663 a(n) = (5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

Original entry on oeis.org

1, 1, 8, 208, 12480, 1435200, 281299200, 86640153600, 39507910041600, 25482601976832000, 22424689739612160000, 26147188236387778560000, 39429959860472770068480000, 75350653293363463600865280000, 179334554838205043370059366400000, 523656900127558726640573349888000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 16 2016

Keywords

Comments

Heptagonal pyramidal factorial numbers.

Links

Crossrefs

Cf. A002413.
Cf. A084940 (heptagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279662 (hexagonal pyramidal factorial numbers).

Programs

  • Magma
    [Round((5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5)): n in [0..20]]; // Vincenzo Librandi Dec 17 2016
  • Mathematica
    FullSimplify[Table[(5/6)^n Gamma[n + 3/5] Gamma[n + 1] Gamma[n + 2]/Gamma[3/5], {n, 0, 15}]]

Formula

a(n) = Product_{k=1..n} k*(k + 1)*(5*k - 2)/6, a(0)=1.
a(n) = Product_{k=1..n} A002413(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(5/6)^n*n^(3*n+21/10)/(Gamma(3/5)*exp(3*n)).
Showing 1-3 of 3 results.

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