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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A091811 Array read by rows: T(n,k) = binomial(n+k-2,k-1)*binomial(2*n-1,n-k).

Original entry on oeis.org

1, 3, 2, 10, 15, 6, 35, 84, 70, 20, 126, 420, 540, 315, 70, 462, 1980, 3465, 3080, 1386, 252, 1716, 9009, 20020, 24024, 16380, 6006, 924, 6435, 40040, 108108, 163800, 150150, 83160, 25740, 3432, 24310, 175032, 556920, 1021020, 1178100, 875160
Offset: 1

Views

Author

Benoit Cloitre, Mar 18 2004

Keywords

Comments

Alternating sum of elements of n-th row = 1.
If a certain event has a probability p of occurring in any given trial, the probability of its occurring at least n times in 2n-1 trials is Sum_{k=1..n} T(n,k)*(-1)^(k-1)*p^(n+k-1). For example, the probability of its occurring at least 4 out of 7 times is 35p^4 - 84p^5 + 70p^6 - 20p^7. - Matthew Vandermast, Jun 05 2004
With the row polynomial defined as R(n,x) = Sum_{k = 1..n} T(n,k)*x^k, the row polynomial is related to the regularized incomplete Beta function I_x(a,b), through the relation R(n,x) = -(-x)^{-n+1}*I_{-x}(n,n). - Leo C. Stein, Jun 06 2019

Examples

			Triangle starts:
    1,
    3,   2,
   10,  15,   6,
   35,  84,  70,  20,
  126, 420, 540, 315, 70,
  ...

Crossrefs

Cf. A001700 (first column), A002740 (second column), A000984 (main diagonal), A033876 (second diagonal), A178792 (row sums).

Programs

  • Magma
    [[Binomial(n+k-2,k-1)*Binomial(2*n-1,n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 15 2015
  • Mathematica
    t[n_, k_] := Binomial[n+k-2, k-1]*Binomial[2n-1, n-k]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 06 2012 *)
  • PARI
    T(x,y)=binomial(x+y-2,y-1)*binomial(2*x-1,x-y)

Formula

From Peter Bala, Apr 10 2012: (Start)
O.g.f.: x*t*(1+2*x-sqrt(1-4*t*(x+1)))/(2*(x+t)*sqrt(1-4*t*(x+1))) = x*t + (3*x+2*x^2)*t^2 + (10*x+15*x^2+6*x^3)*t^3 + ....
Sum_{k = 1..n} (-1)^(k-1)*T(n,k)*2^(n-k) = 4^(n-1).
Row polynomial R(n+1,x) = ((2*n+1)!/n!^2)*x*Integral_{y = 0..1} (y*(1+x*y))^n dy. Row sums A178792. (End)

A188289 Binomial sum related to rooted trees.

Original entry on oeis.org

0, 2, 3, 14, 45, 167, 609, 2270, 8517, 32207, 122463, 467843, 1794195, 6903353, 26635773, 103020254, 399300165, 1550554583, 6031074183, 23493410759, 91638191235, 357874310213, 1399137067683, 5475504511859, 21447950506395, 84083979575117
Offset: 0

Views

Author

Olivier Gérard, Aug 19 2012

Keywords

Links

Crossrefs

Cf. A000984, A014300, A026641, A178792, A176479, A072547.

Programs

  • GAP
    List([0..30], n-> Binomial(2*n,n) -(-1)^n -Sum([0..n-1], k-> Binomial(2*k,n-1))); # G. C. Greubel, Apr 29 2019
  • Magma
    [n eq 0 select 0 else Binomial(2*n, n) -(-1)^n - (&+[Binomial(2*k, n-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Apr 29 2019
  • Mathematica
    Table[Binomial[2n,n]-(-1)^n-Sum[Binomial[2k,n-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Dec 10 2012 *)
  • PARI
    {a(n) = binomial(2*n,n) -(-1)^n -sum(k=0,n-1, binomial(2*k,n-1))}; \\ G. C. Greubel, Apr 29 2019
  • Sage
    [binomial(2*n,n) -(-1)^n -sum(binomial(2*k, n-1) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Apr 29 2019

Formula

a(n) = binomial(2*n,n) - (-1)^n - Sum_{k=0..n-1} binomial(2*k, n-1).
a(n) = Sum_{k=1..n} binomial(n+k,k)*(Sum_{r=n-k..n} (-1)^r*binomial(n-k, r)).
a(n) = (-1)^n*2^(-(1+n))*(1 - 2^(1+n) + (-2)^n*binomial(2+2*n, 1+n) * hypergeometric2F1(1, 2+2*n; 2+n; -1)).

A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Cf. A098430, A383716, A386811.
Cf. A178792, A244038, A386895.

Programs

  • Mathematica
    Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025
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