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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.
7

%I #9 Feb 07 2014 16:16:50

%S 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,

%T 28,137,210,157,52,12,1,1,36,245,528,535,264,81,11,1,1,45,414,1219,

%U 1622,1086,425,97,15,1,1,55,669,2621,4494,3921,1965,626,129,14,1,1,66,1042

%N 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.

%H Paul D. Hanna, <a href="/A162430/b162430.txt">Table of n, a(n) for n = 1..1275</a>

%e This table begins:

%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

%e 1,3,4,7,7,9,12,11,15,14,19,19,19,22,23,27,26,27,31,30,33,35,35,40,...

%e 1,6,13,25,39,52,81,97,129,154,187,234,250,321,337,406,468,493,579,...

%e 1,10,33,76,157,264,425,626,897,1230,1629,2174,2653,3448,4119,4978,...

%e 1,15,71,210,535,1086,1965,3431,5425,8181,12165,17211,23345,31980,...

%e 1,21,137,528,1622,3921,8254,16396,29136,48773,79307,121743,180415,...

%e 1,28,245,1219,4494,12936,31767,70826,141891,264131,468482,785401,...

%e 1,36,414,2621,11602,39622,112951,283574,637706,1318351,2557686,...

%e 1,45,669,5317,28275,113922,375337,1064274,2679558,6142420,...

%e 1,55,1042,10280,65601,310314,1177530,3774455,10626160,26954099,...

%e 1,66,1573,19085,145751,806465,3514434,12733216,40034302,...

%e 1,78,2311,34211,311524,2010329,10036832,41072816,144045962,...

%e ...

%e Let coefficients in powers of the series:

%e S = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...

%e form the following sequences:

%e S^1: [(1),(1,0),(1,0,0),(1,0,0,0),(1,0,0,0,0),(1,0,0,0,0,0),...]

%e S^2: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),(2,4,0,2,0,1),...]

%e S^3: [(1),(3,3),(4,6,3),(6,9,3,7),(9,6,9,9,6),(6,15,9,7,12,3),...]

%e S^4: [(1),(4,6),(8,13,12),(14,24,18,20),(32,24,31,40,30),...]

%e S^5: [(1),(5,10),(15,25,31),(35,55,60,60),(90,90,95,135,125),...]

%e S^6: [(1),(6,15),(26,45,66),(82,120,156,170),(231,276,290,390,...]

%e S^7: [(1),(7,21),(42,77,126),(175,253,357,434),(567,735,833,...]

%e S^8: [(1),(8,28),(64,126,224),(344,512,757,1008),(1332,1792,...]

%e S^9: [(1),(9,36),(93,198,378),(633,990,1521,2173),(2979,4113,...]

%e S^10:[(1),(10,45),(130,300,612),(1105,1830,2925,4420),(6341,...]

%e ...

%e then the sums of the above grouped terms (enclosed in parenthesis)

%e form the initial terms of the rows of this table. Examples:

%e T(3,4) = (9+6+9+9+6) = 39 ;

%e T(4,3) = (14+24+18+20) = 76 ;

%e T(5,3) = (35+55+60+60) = 210.

%e Summing the coefficients of S^n in this way generates all the rows of this table.

%t 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 *)

%o (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))}

%o for(n=1,12,for(k=0,12,print1(T(n,k),", "));print(""))

%Y Cf. rows: A162431, A162432, A162433.

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

%Y Cf. A162424 (variant).

%K nonn,tabl

%O 1,5

%A _Paul D. Hanna_, Jul 03 2009