A127082 Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recursion: C_k(x) = ( 1 + Sum_{n>=1} x^n*C_{n-1+k}(x) )^(k+1).

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 28, 15, 4, 1, 64, 127, 85, 26, 5, 1, 308, 650, 531, 192, 40, 6, 1, 1728, 3737, 3600, 1551, 365, 57, 7, 1, 11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1, 79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1

Offset

0,4

Comments

This is a variant of triangle A124328.

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Example

C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;
C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;
C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;
C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;
thus the row sums equal column 0 shift left.
The triangle begins:
       1;
       1,       1;
       2,       2,       1;
       5,       7,       3,       1;
      16,      28,      15,       4,      1;
      64,     127,      85,      26,      5,     1;
     308,     650,     531,     192,     40,     6,     1;
    1728,    3737,    3600,    1551,    365,    57,     7,    1;
   11046,   23996,   26266,   13416,   3635,   620,    77,    8,   1;
   79065,  170866,  205353,  122770,  38556,  7356,   973,  100,   9,  1;
  625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440, 126, 10, 1;

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c, k, r}], {r, k, n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 30 2020 *)

Prog

(PARI) {T(n, k)=if(n==k, 1, polcoeff( (1 + x*sum(r=k, n-1, x^(r-k)*sum(c=k, r, T(r, c) ))+x*O(x^n))^(k+1), n-k))}

Crossrefs

Cf. variant: A124328;

Columns: A127083, A127084, A127085, A127086, A127090 (central terms).

Sequence in context: A309991 A162382 A325580 * A297628 A342722 A344528

Adjacent sequences: A127079 A127080 A127081 * A127083 A127084 A127085

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 04 2007

Status

approved