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 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. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified August 2 21:08 EDT 2024. Contains 374875 sequences. (Running on oeis4.)