A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).
1, 1, 1, 2, 4, 1, 9, 21, 9, 1, 64, 156, 84, 16, 1, 630, 1540, 935, 230, 25, 1, 7916, 19160, 12480, 3564, 510, 36, 1, 121023, 288813, 196623, 61845, 10465, 987, 49, 1, 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1, 45179508, 104657520
Offset: 0
Examples
Triangle begins: 1; 1, 1; 2, 4, 1; 9, 21, 9, 1; 64, 156, 84, 16, 1; 630, 1540, 935, 230, 25, 1; 7916, 19160, 12480, 3564, 510, 36, 1; 121023, 288813, 196623, 61845, 10465, 987, 49, 1; 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1; 45179508, 104657520, 74847168, 26415840, 5426949, 695079, 56511, 2844, 81, 1; 1059312264, 2420186616, 1755406674, 642448632, 140247810, 19683060, 1830080, 112520, 4410, 100, 1; ... Coefficients in self-compositions 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 rows of A122888 into diagonals of A122888; the initial diagonals begin: A112319: [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...]; A112317: [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028,..]; A112320: [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...]. For example: T * [1, 0, 0, 0, 0, 0, 0,...]~ = A112319; T * [1, 1, 0, 0, 0, 0, 0,...]~ = A112317; T * [1, 2, 2, 1, 0, 0, 0,...]~ = A112320.
Links
- Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)
Programs
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PARI
{T(n, k)=local(F=x, M, N, P, m=n); 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, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
Comments