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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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A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 8, 3, 5, 69, 77, 15, 14, 692, 1749, 890, 105, 42, 8120, 41998, 41909, 12039, 945, 132, 110278, 1114808, 1944225, 1018865, 186594, 10395, 429, 1707965, 33058519, 94833341, 80595226, 25798856, 3260067, 135135, 1430, 29750636, 1093994697, 4979407614, 6439957299, 3201618970, 687652446, 63390060, 2027025
Offset: 0

Views

Author

Alois P. Heinz, May 23 2015

Keywords

Examples

			Triangle T(n,k) begins:
:   1;
:   1,      1;
:   2,      8,       3;
:   5,     69,      77,      15;
:  14,    692,    1749,     890,     105;
:  42,   8120,   41998,   41909,   12039,    945;
: 132, 110278, 1114808, 1944225, 1018865, 186594, 10395;

Links

Crossrefs

Column k=0 gives A000108.
Main diagonal gives A001147.
Row sums give A258224.
T(2n,n) gives A292695.
Cf. A258220, A258222.

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:= (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, (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!;
Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i).

A258221 Row sums of A258220.

Original entry on oeis.org

1, 2, 11, 90, 952, 12203, 182677, 3118314, 59688447, 1265193199, 29408221404, 743677646836, 20325564686926, 597051775012306, 18758388926380409, 627712133246362442, 22288938527631882996, 837033514431748421053, 33146037056721682537319, 1380365444443138768970878
Offset: 0

Views

Author

Alois P. Heinz, May 23 2015

Keywords

Links

Crossrefs

Cf. A258220, A258224.

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:= 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..25);
  • 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!;
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} A258220(n,k).
Showing 1-2 of 2 results.

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