A162430 Square table defined by T(n,k) = Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^n for n>=1, k>=0, where S(x) = Sum_{n>=0} x^(n(n+1)/2), as read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 13, 7, 1, 1, 15, 33, 25, 7, 1, 1, 21, 71, 76, 39, 9, 1, 1, 28, 137, 210, 157, 52, 12, 1, 1, 36, 245, 528, 535, 264, 81, 11, 1, 1, 45, 414, 1219, 1622, 1086, 425, 97, 15, 1, 1, 55, 669, 2621, 4494, 3921, 1965, 626, 129, 14, 1, 1, 66, 1042

Offset

1,5

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1275

Example

This table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,4,7,7,9,12,11,15,14,19,19,19,22,23,27,26,27,31,30,33,35,35,40,...
1,6,13,25,39,52,81,97,129,154,187,234,250,321,337,406,468,493,579,...
1,10,33,76,157,264,425,626,897,1230,1629,2174,2653,3448,4119,4978,...
1,15,71,210,535,1086,1965,3431,5425,8181,12165,17211,23345,31980,...
1,21,137,528,1622,3921,8254,16396,29136,48773,79307,121743,180415,...
1,28,245,1219,4494,12936,31767,70826,141891,264131,468482,785401,...
1,36,414,2621,11602,39622,112951,283574,637706,1318351,2557686,...
1,45,669,5317,28275,113922,375337,1064274,2679558,6142420,...
1,55,1042,10280,65601,310314,1177530,3774455,10626160,26954099,...
1,66,1573,19085,145751,806465,3514434,12733216,40034302,...
1,78,2311,34211,311524,2010329,10036832,41072816,144045962,...
...
Let coefficients in powers of the series:
S = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
form the following sequences:
S^1: [(1),(1,0),(1,0,0),(1,0,0,0),(1,0,0,0,0),(1,0,0,0,0,0),...]
S^2: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),(2,4,0,2,0,1),...]
S^3: [(1),(3,3),(4,6,3),(6,9,3,7),(9,6,9,9,6),(6,15,9,7,12,3),...]
S^4: [(1),(4,6),(8,13,12),(14,24,18,20),(32,24,31,40,30),...]
S^5: [(1),(5,10),(15,25,31),(35,55,60,60),(90,90,95,135,125),...]
S^6: [(1),(6,15),(26,45,66),(82,120,156,170),(231,276,290,390,...]
S^7: [(1),(7,21),(42,77,126),(175,253,357,434),(567,735,833,...]
S^8: [(1),(8,28),(64,126,224),(344,512,757,1008),(1332,1792,...]
S^9: [(1),(9,36),(93,198,378),(633,990,1521,2173),(2979,4113,...]
S^10:[(1),(10,45),(130,300,612),(1105,1830,2925,4420),(6341,...]
...
then the sums of the above grouped terms (enclosed in parenthesis)
form the initial terms of the rows of this table. Examples:
T(3,4) = (9+6+9+9+6) = 39 ;
T(4,3) = (14+24+18+20) = 76 ;
T(5,3) = (35+55+60+60) = 210.
Summing the coefficients of S^n in this way generates all the rows of this table.

Mathematica

t[n_, k_] := Module[{s = Sum[x^(m*(m+1)/2), {m, 0, k+1}]+O[x]^((k+1)*(k+2)/2)}, Sum[Coefficient[s^n, x, m], {m, k*(k+1)/2, k*(k+1)/2+k}]]; Table[t[n-k+1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 18 2013 *)

Prog

(PARI) {T(n, k)=local(S=sum(m=0, k+1, x^(m*(m+1)/2))+O(x^((k+1)*(k+2)/2))); sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^n, m))}
for(n=1, 12, for(k=0, 12, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. rows: A162431, A162432, A162433.

Cf. A162434 (antidiagonal sums), A162435 (main diagonal).

Cf. A162424 (variant).

Sequence in context: A184049 A125230 A208334 * A305059 A355996 A128101

Adjacent sequences: A162427 A162428 A162429 * A162431 A162432 A162433

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 03 2009

Status

approved