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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A116382 Riordan array (1/sqrt(1-4*x^2), (1-2*x^2*c(x^2))*(x^2*c(x^2))/(x*(1-x-x^2*c(x^2)))) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 6, 4, 5, 3, 1, 0, 10, 10, 8, 4, 1, 20, 15, 21, 19, 12, 5, 1, 0, 35, 42, 42, 32, 17, 6, 1, 70, 56, 84, 92, 77, 50, 23, 7, 1, 0, 126, 168, 192, 180, 131, 74, 30, 8, 1, 252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

Row sums are A116383. Diagonal sums are A116384.
First column has e.g.f. Bessel_I(0,2*x) (A000984 with interpolated zeros).
Second column has e.g.f. Bessel_I(1,2*x) + Bessel_I(2,2*x) (A037952).
Third column has e.g.f. Bessel_I(2,2*x) + 2*Bessel_I(3,2*x) + Bessel_I(4,2*x) (A116385).
A binomial-Bessel triangle: column k has e.g.f. Sum_{j=0..k} C(k,j) * Bessel_I(k+j,2*x).

Examples

			Triangle begins
    1;
    0,   1;
    2,   1,   1;
    0,   3,   2,   1;
    6,   4,   5,   3,   1;
    0,  10,  10,   8,   4,   1;
   20,  15,  21,  19,  12,   5,   1;
    0,  35,  42,  42,  32,  17,   6,   1;
   70,  56,  84,  92,  77,  50,  23,   7,  1;
    0, 126, 168, 192, 180, 131,  74,  30,  8, 1;
  252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1;

Links

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m)  ))))); # G. C. Greubel, May 22 2019
  • Magma
    T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
[[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 22 2019
  • Mathematica
    T[n_, k_] := Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2018 *)
  • PARI
    {T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))}; \\ G. C. Greubel, May 22 2019
  • Sage
    def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 22 2019

Formula

Riordan array (1/sqrt(1-4*x^2), sqrt(1-4*x^2)*(1-sqrt(1-4*x^2))/(x-2*x^2 + x*sqrt(1-4*x^2))).
Number triangle T(n,k) = Sum{j=0..n} (-1)^(n-j)* C(n,j)*Sum_{i=0..j} C(j,i-k)*C(i,j-i).

A051631 Triangle formed using Pascal's rule except begin and end n-th row with n-1.

Original entry on oeis.org

-1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 3, 3, 2, 3, 3, 4, 6, 5, 5, 6, 4, 5, 10, 11, 10, 11, 10, 5, 6, 15, 21, 21, 21, 21, 15, 6, 7, 21, 36, 42, 42, 42, 36, 21, 7, 8, 28, 57, 78, 84, 84, 78, 57, 28, 8, 9, 36, 85, 135, 162, 168, 162, 135, 85, 36, 9
Offset: 0

Views

Author

Asher Auel

Keywords

Comments

Row sums give A000918(n).
Central terms for n>0: T(2*n,n)=A024483(n+1), T(n,[n/2])=A116385(n-1); for n>1: T(n,1) = T(n,n-1) = A000217(n-2). - Reinhard Zumkeller, Nov 13 2011

Examples

			Triangle begins
  -1;
   0, 0;
   1, 0, 1;
   2, 1, 1, 2;
   3, 3, 2, 3, 3;
   4, 6, 5, 5, 6, 4; ...

Links

Crossrefs

Cf. A007318.

Programs

  • Haskell
    a051631 n k = a051631_tabl !! n !! k
a051631_row n = a051631_tabl !! n
a051631_list = concat a051631_tabl
a051631_tabl = iterate (\row -> zipWith (+) ([1] ++ row) (row ++[1])) [-1]
-- Reinhard Zumkeller, Nov 13 2011
  • Magma
    /* As triangle */ [[Binomial(n+2,k+1) - 3*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 11 2019
  • Mathematica
    Clear[t]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; t[n_, 0] := n-1; t[n_, n_] := n-1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 11 2013 *)

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n, T(n,0) = T(n,n) = n - 1.
T(n,k) = C(n+2,k+1) - 3*C(n,k). - Charlie Neder, Jan 10 2019

Extensions

Definition modified and keyword tabl added by Reinhard Zumkeller, Nov 13 2011
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