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.

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A248326 Square array read by downward antidiagonals: super Patalan numbers of order 5.

Original entry on oeis.org

1, 5, 20, 75, 50, 450, 1375, 500, 750, 10500, 27500, 6875, 5625, 13125, 249375, 577500, 110000, 61875, 78750, 249375, 5985000, 12512500, 1925000, 825000, 721875, 1246875, 4987500, 144637500, 277062500, 35750000, 12375000, 8250000, 9796875, 21375000, 103312500, 3512625000, 6233906250, 692656250
Offset: 0

Views

Author

Tom Richardson, Oct 04 2014

Keywords

Comments

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 5, A025750.

Examples

			T(0..4,0..4) is
  1       5       75      1375    27500
  20      50      500     6875    110000
  450     750     5625    61875   825000
  10500   13125   78750   721875  8250000
  249375  249375  1246875 9796875 97968750

Links

Crossrefs

Cf. A068555, A025750, A034688 (first row), A049382 (first column), A248324, A248325, A248328, A248329, A248332.

Programs

  • PARI
    matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*25^(n+k)*binomial(n-1/5,n+k)) \\ Michel Marcus, Oct 09 2014

Formula

T(0,0)=1, T(n,k) = T(n-1,k)*(25*n-5)/(n+k), T(n,k) = T(n,k-1)*(25*k-20)/(n+k).
G.f.: (x/(1-25*x)^(4/5)+y/(1-25*y)^(1/5))/(x+y-25*x*y).
T(n,k) = (-1)^k*25^(n+k)*binomial(n-1/5,n+k).

A049397 Expansion of (1-25*x)^(-9/5).

Original entry on oeis.org

1, 45, 1575, 49875, 1496250, 43391250, 1229418750, 34248093750, 941822578125, 25638503515625, 692239594921875, 18564607318359375, 495056195156250000, 13138029794531250000, 347219358855468750000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-25*x)^(-9/5) = 1 + 9/5*(5^2*x) + 63/25*(5^2*x)^2 + 399/125*(5^2*x)^3 + ... = 1 + 5*9*x + 63*5^2* x^2 + 399*5^3*x^3 + ... = 1 + 45*x + 1575*x^2 + 49875*x^3 + ...

Links

Crossrefs

Cf. A049382.

Programs

  • Mathematica
    a[n_Integer/;n>=0]:=25^n Pochhammer[9/5,n]/n! (* Todd Silvestri, Dec 17 2014 *)

Formula

G.f.: (1-25*x)^(-9/5).
a(n) = 5^n/n! * product[ k=0..n-1 ] (5*k+9).
a(n) ~ 5/4*Gamma(4/5)^-1*n^(4/5)*5^(2*n)*{1 + 18/25*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = (25^n*(9/5)n)/n!, where the rising factorial (c)_n = Gamma(c+n)/Gamma(c). - _Todd Silvestri, Dec 17 2014. See the a(n) formula above.

Extensions

Edited: name and example corrected according to G.f. - Wolfdieter Lang, Jan 05 2015

A216787 a(n) = Product_{k=1..n} (144 - 12/k).

Original entry on oeis.org

1, 132, 18216, 2550240, 359583840, 50917071744, 7230224187648, 1028757612985344, 146597959850411520, 20914642271992043520, 2986610916440463814656, 426813850967673556058112, 61034380688377318516310016, 8732611390798600956948971520, 1250010944797171165551838494720
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706, A216786.

Programs

  • Maple
    seq(product(144-12/k, k=1.. n), n=0..20);
seq((12^n/n!)*product(12*k+11, k=0.. n-1), n=0..20);
  • Mathematica
    Join[{1},FoldList[Times,144-12/Range[20]]] (* Harvey P. Dale, Dec 22 2015 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 144^n * Gamma(n+11/12) / (Gamma(11/12) * Gamma(n+1)).
a(n) ~ c * 144^n / n^(1/12), where c = 1/Gamma(11/12) = 0.947376... . (End)

A216788 a(n) = Product_{k=1..n} (169 - 13/k).

Original entry on oeis.org

1, 156, 25350, 4174300, 691890225, 115130533440, 19207610662240, 3210414924974400, 537343198067590200, 90034838076214002400, 15098842345381088202480, 2533860269961226256525280, 425477370330989242241536600, 71480198215606192696578148800
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706, A216786, A216787.

Programs

  • Maple
    seq(product(169-13/k, k=1.. n), n=0..20);
seq((13^n/n!)*product(13*k+12, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[169-13/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Mar 13 2013 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 169^n * Gamma(n+12/13) / (Gamma(12/13) * Gamma(n+1)).
a(n) ~ c * 169^n / n^(1/13), where c = 1/Gamma(12/13) = 0.951742... . (End)
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