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 A162434 Antidiagonal sums of table A162430. 7
 1, 2, 5, 12, 32, 82, 218, 598, 1702, 4925, 14496, 43720, 134854, 423988, 1357884, 4425384, 14664610, 49386081, 168884638, 586012116, 2061993475, 7353614415, 26566226591, 97177708497, 359765742126, 1347447717234, 5103618615825 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..60 FORMULA a(n) = Sum_{k=0..n} Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^(n-k) for n>=1 where S(x) = Sum_{n>=0} x^(n(n+1)/2). EXAMPLE 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),...] S^2: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),...] S^3: [(1),(3,3),(4,6,3),(6,9,3,7),(9,6,9,9,6),...] S^4: [(1),(4,6),(8,13,12),(14,24,18,20),...] S^5: [(1),(5,10),(15,25,31),(35,55,60,60),...] S^6: [(1),(6,15),(26,45,66),(82,120,156,170),...] ... then the sums of the above grouped terms (in parenthesis) form the initial terms of the rows of table A162430: 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,... 1,6,13,25,39,52,81,97,129,154,187,234,250,321,... 1,10,33,76,157,264,425,626,897,1230,1629,2174,... 1,15,71,210,535,1086,1965,3431,5425,8181,12165,... 1,21,137,528,1622,3921,8254,16396,29136,48773,... 1,28,245,1219,4494,12936,31767,70826,141891,... 1,36,414,2621,11602,39622,112951,283574,637706,... ... The antidiagonal sums of the above table forms this sequence. 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[Sum[t[n-k+1, k], {k, 0, n}], {n, 0, 26}] (* Jean-François Alcover, Nov 18 2013 *) PROG (PARI) {a(n)=local(S=sum(m=0, n+1, x^(m*(m+1)/2))+O(x^((n+1)*(n+2)/2))); sum(k=0, n, sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^(n-k), m)))} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A162430, A162431, A162432, A162433, A162435, A162428 (variant). Sequence in context: A232215 A265265 A293868 * A277863 A039809 A335456 Adjacent sequences: A162431 A162432 A162433 * A162435 A162436 A162437 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 03 2009 STATUS approved

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Last modified August 15 02:02 EDT 2024. Contains 375171 sequences. (Running on oeis4.)