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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).
11
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 04 2007
STATUS
approved