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.

A106188 Expansion of 1/((1-x^2)*sqrt(1-4*x)).

Original entry on oeis.org

1, 2, 7, 22, 77, 274, 1001, 3706, 13871, 52326, 198627, 757758, 2902783, 11158358, 43019383, 166275878, 644099773, 2499882098, 9719235073, 37845145898, 147565763893, 576103020338, 2251664727613, 8809533747938, 34499268410713
Offset: 0

Views

Author

Paul Barry, Apr 24 2005

Keywords

Comments

Diagonal sums of number triangle A106187.

Examples

			1 + 2*x + 7*x^2 + 22*x^3 + 77*x^4 + 274*x^5 + 1001*x^6 + 3706*x^7 + 13871*x^8 + ...

Links

Crossrefs

Cf. A006134, A054108. Convolution of A000984 and A059841.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-x^2)*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)
  • PARI
    x='x+O('x^50); Vec(1/((1-x^2)*sqrt(1-4*x))) \\ G. C. Greubel, Mar 16 2017

Formula

G.f. 1 / ((1 - x^2) * sqrt(1 - 4*x)).
a(n)=sum{k=0..floor(n/2), binomial(2(n-2k), n-2k)}.
PSUMSIGN transform of A006134. a(n+1) + a(n) = A006134(n). a(n) = Sum_{k=0..n} (-1)^k * binomial(2 * (n-k), n-k). - Michael Somos, Jun 20 2012
First difference is A054108. a(n+1) - a(n) = A054108(n). - Michael Somos, Jun 20 2012
D-finite with recurrence: n*a(n)+2*(1-2*n)*a(n-1) -n*a(n-2) +2*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 09 2012
a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 03 2014

A106190 Triangle read by rows: T(n,k) = binomial(2(n-k),n-k)/(1-2(n-k)).

Original entry on oeis.org

1, -2, 1, -2, -2, 1, -4, -2, -2, 1, -10, -4, -2, -2, 1, -28, -10, -4, -2, -2, 1, -84, -28, -10, -4, -2, -2, 1, -264, -84, -28, -10, -4, -2, -2, 1, -858, -264, -84, -28, -10, -4, -2, -2, 1, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -9724, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -33592, -9724, -2860, -858
Offset: 0

Views

Author

Paul Barry, Apr 24 2005

Keywords

Comments

Sequence array for expansion of sqrt(1-4x).
Row sums are A106191. Diagonal sums are A106192. Sequence array for A002420. Inverse of number triangle A106187.
Riordan array (sqrt(1-4x),x).

Examples

			Triangle begins
1;
-2,1;
-2,-2,1;
-4,-2,-2,1;
-10,-4,-2,-2,1;
-28,-10,-4,-2,-2,1;

Programs

  • Mathematica
    T[n_, k_] := Binomial[2(n - k), n - k]/(1 - 2(n - k)); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Apr 25 2005 *)

Extensions

More terms from Robert G. Wilson v, Apr 25 2005

A382779 a(n) = Sum_{0<=i<=k<=n} 2^(4*(n-k)) * binomial(2*i,i)^2 * binomial(2*n-2*i,n-i) * binomial(2*k-2*i,k-i) * binomial(2*k,k)^2 * binomial(2*n-2*k,n-k).

Original entry on oeis.org

1, 96, 14944, 2743296, 547115616, 114691716096, 24855999978496, 5516395226824704, 1246310097807086176, 285511424277840331776, 66136775263705972306944, 15459962390271174936920064, 3641349843333453310791883776, 863175698505287814277639471104, 205741271729612742942836920909824
Offset: 0

Views

Author

Stefano Spezia, May 11 2025

Keywords

Links

Crossrefs

Cf. A000984, A001025, A002894, A106187.

Programs

  • Mathematica
    a[n_]:=Sum[Sum[2^(4(n-k))Binomial[2i,i]^2Binomial[2n-2i,n-i]Binomial[2k-2i,k-i]Binomial[2k,k]^2Binomial[2n-2k,n-k],{i,0,n}],{k,0,n}]; Array[a,15,0]

Formula

Recurrence: (n + 1)^5 * a(n+1) - 32 * (2*n + 1) * (8*n^4 + 16*n^3 + 20*n^2 + 12*n + 3) * a(n) + 2^16 * n^5 * a(n-1) = 0 (see Lai et al., p. 2).
a(n) = Sum_{k=0..n} 2^(4*(n-k)) * binomial(2*k,k)^3 * binomial(2*n,n) * binomial(2*n-2*k,n-k) * hypergeom([1/2, 1/2, -k, -n], [1, 1/2-k, 1/2-n], 1).
Showing 1-3 of 3 results.

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