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 A304188 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)*(n+2)) / A(x) = 0 for n>0. 3
 1, 6, 30, 264, 4179, 97758, 3000084, 113020056, 5018695542, 255724146876, 14671199172480, 934467807541824, 65366076594301044, 4978197982191048600, 409875168025688997456, 36268233577292228677728, 3431775207222740657912472, 345742547371677388835049744, 36948141363745699171977916032, 4174429749114285739841190548928 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..19. FORMULA A132613(n+1) = [x^n] (1+x)^((n+2)*(n+3)) / A(x) for n>0. EXAMPLE G.f.: A(x) = 1 + 6*x + 30*x^2 + 264*x^3 + 4179*x^4 + 97758*x^5 + 3000084*x^6 + 113020056*x^7 + 5018695542*x^8 + 255724146876*x^9 + 14671199172480*x^10 + ... ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in (1+x)^((n+1)*(n+2)) / A(x) begins: n=0: [1, -4, -5, -114, -2289, -62568, -2113983, -84889290, ...]; n=1: [1, 0, -15, -154, -2790, -72432, -2378450, -93729900, ...]; n=2: [1, 6, 0, -224, -3924, -91776, -2858196, -109145280, ...]; n=3: [1, 14, 76, 0, -5310, -128964, -3714456, -134815824, ...]; n=4: [1, 24, 261, 1510, 0, -169752, -5223348, -178378752, ...]; n=5: [1, 36, 615, 6446, 41121, 0, -6779045, -251285430, ...]; n=6: [1, 50, 1210, 18696, 201435, 1424178, 0, -323428800, ...]; n=7: [1, 66, 2130, 44616, 675591, 7663626, 59857416, 0, ...]; ... in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)*(n+2)) / A(x) = 0 for n>0. RELATED SEQUENCES. The secondary diagonal in the above table that begins [1, 6, 76, 1510, 41121, 1424178, 59857416, 2957282370, ...] yields A132613, column 2 of triangle A132610. Related triangular matrix T = A132610 begins: 1; 1, 1; 2, 1, 1; 14, 4, 1, 1; 194, 39, 6, 1, 1; 4114, 648, 76, 8, 1, 1; 118042, 15465, 1510, 125, 10, 1, 1; 4274612, 483240, 41121, 2908, 186, 12, 1, 1; 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ... in which row n+1 of T = row n of matrix power T^(2*n) with appended '1' for n>=0. PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m+1))/Ser(A) )[m] ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A132613, A304189, A304192, A304193, A132610. Sequence in context: A270845 A277073 A052585 * A343574 A051821 A099031 Adjacent sequences: A304185 A304186 A304187 * A304189 A304190 A304191 KEYWORD nonn AUTHOR Paul D. Hanna, May 09 2018 STATUS approved

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Last modified August 13 11:56 EDT 2024. Contains 375138 sequences. (Running on oeis4.)