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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A258220 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 6, 1, 25, 49, 15, 1, 208, 498, 217, 28, 1, 2146, 6016, 3360, 635, 45, 1, 26368, 84042, 56728, 13997, 1475, 66, 1, 375733, 1332661, 1046619, 316281, 43974, 2954, 91, 1, 6092032, 23660034, 21053089, 7479444, 1283817, 114576, 5334, 120, 1
Offset: 0

Views

Author

Alois P. Heinz, May 23 2015

Keywords

Examples

			Triangle T(n,k) begins:
:     1;
:     1,     1;
:     4,     6,     1;
:    25,    49,    15,     1;
:   208,   498,   217,    28,    1;
:  2146,  6016,  3360,   635,   45,  1;
: 26368, 84042, 56728, 13997, 1475, 66, 1;

Links

Crossrefs

Column k=0 gives A005411 (for n>0).
Main diagonal and lower diagonal give: A000012, A000384(n+1).
Row sums give A258221.
T(2n,n) gives A292692.
Cf. A258219, A258223.

Programs

  • Maple
    b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    b[x_, y_, t_, k_] := b[x, y, t, k] = If[y>x || y<0, 0, If[x==0, 1, b[x-1, y - 1, False, k]*If[t, (x+k*y)/y, 1] + b[x-1, y+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; T [n_, k_] := Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/k!; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i).

A258224 Row sums of A258223.

Original entry on oeis.org

1, 2, 13, 166, 3450, 105053, 4385297, 239389538, 16497800177, 1396841773631, 142194450687440, 17100401655609460, 2394468068218870494, 385647096554809325098, 70702689662684594772871, 14623755150209185924416598, 3385915623744083331349813602
Offset: 0

Views

Author

Alois P. Heinz, May 23 2015

Keywords

Links

Crossrefs

Cf. A258221, A258223.

Programs

  • Maple
    b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= proc(n,k) option remember;
       add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
    end:
a:= proc(n) option remember; add(T(n,k), k=0..n) end:
seq(a(n), n=0..20);
  • Mathematica
    b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0,
     If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1]
                 + b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[2*n, 0, False, k];
T[n_, k_] := Sum[A[n, i]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} A258223(n,k).
Showing 1-2 of 2 results.

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