A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).

1, 1, 1, 2, 2, 1, 8, 7, 3, 1, 50, 40, 15, 4, 1, 436, 326, 112, 26, 5, 1, 4912, 3492, 1128, 240, 40, 6, 1, 68098, 46558, 14373, 2881, 440, 57, 7, 1, 1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1, 21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Formula

Columns may be generated by a method illustrated by triangles A187005, A187115, and A187120. The main diagonal of triangles A187005, A187115, and A187120, equals columns 0, 1, and 2, respectively.

Example

Triangle begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1;
68098, 46558, 14373, 2881, 440, 57, 7, 1;
1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1;
21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100, 9, 1; ...
Coefficients in iterations of (x+x^2) form table A122888:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
This triangle T transforms one diagonal in the above table into another;
start with the main diagonal of A122888, A112319, which begins:
[1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
then the transform T*A112319 equals A112317, which begins:
[1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, ...];
and the transform T*A112317 equals A112320, which begins:
[1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].

Prog

(PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
(PARI) /* Generate by method given in A187005, A187115, A187120 (faster): */
{T(n, k)=local(Ck=x); for(m=1, n-k+1, Ck=(1/x^k)*subst(truncate(x^k*Ck), x, x+x^2 +x*O(x^m))); polcoeff(Ck, n-k+1, x)}

Crossrefs

Cf. columns: A135081, A135082, A135083.

Cf. related tables: A122888, A166900, A187005, A187115, A187120.

Cf. related sequences: A112319, A112317, A112320, A187009.

Sequence in context: A121350 A339262 A198569 * A348724 A238182 A117260

Adjacent sequences: A135077 A135078 A135079 * A135081 A135082 A135083

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 18 2007

Extensions

Added cross-reference; example corrected and name changed by Paul D. Hanna, Feb 04 2011

Status

approved