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 A220178 Triangle where the g.f. for row n equals  d^n/dx^n (1+x+x^2)^n / n!  for n>=0, as read by rows. 3
 1, 1, 2, 3, 6, 6, 7, 24, 30, 20, 19, 80, 150, 140, 70, 51, 270, 630, 840, 630, 252, 141, 882, 2520, 4200, 4410, 2772, 924, 393, 2856, 9576, 19320, 25410, 22176, 12012, 3432, 1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870, 3139, 29070, 126720, 341880, 630630, 828828, 780780, 514800, 218790, 48620 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Ivan Neretin, Table of n, a(n) for n = 0..5150 (rows 0..100) FORMULA G.f.: A(x,y) = 1 / sqrt(1-2*x-3*x^2 - 4*x*y). G.f.: A(x,y) = Sum_{k>=0} binomial(2*k,k) * x^k*y^k / (1-2*x-3*x^2)^(k+1/2). First column is the central trinomial coefficients (A002426). Main diagonal is the central binomial coefficients (A000984). Row sums form the central coefficients of (1+3*x+3*x^2)^n (A122868). EXAMPLE Triangle begins: 1; 1, 2; 3, 6, 6; 7, 24, 30, 20; 19, 80, 150, 140, 70; 51, 270, 630, 840, 630, 252; 141, 882, 2520, 4200, 4410, 2772, 924; 393, 2856, 9576, 19320, 25410, 22176, 12012, 3432; 1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870; ... The g.f. for column k>=0 equals the central binomial coefficient C(2*k,k) times x^k*y^k*G(x)^(2*k+1) where G(x) = 1/sqrt(1-2*x-3*x^2) is the g.f. of the central trinomial coefficients A002426. The g.f. for row n is d^n/dx^n (1+x+x^2)^n/n!, which begins: n=0: 1; n=1: 1 + 2*x; n=2: 3 + 6*x + 6*x^2; n=3: 7 + 24*x + 30*x^2 + 20*x^3; n=4: 19 + 80*x + 150*x^2 + 140*x^3 + 70*x^4; n=5: 51 + 270*x + 630*x^2 + 840*x^3 + 630*x^4 + 252*x^5; n=6: 141 + 882*x + 2520*x^2 + 4200*x^3 + 4410*x^4 + 2772*x^5 + 924*x^6; ... MATHEMATICA Flatten@Table[CoefficientList[D[(1 + x + x^2)^n/n!, {x, n}], x], {n, 0, 9}] (* Ivan Neretin, Jun 22 2019 *) PROG (PARI) {T(n, k)=polcoeff(polcoeff(1/sqrt(1-2*x-3*x^2 - 4*x*y +x*O(x^n)+y*O(y^k)), n, x), k, y)} for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print("")) (PARI) row(n) = my(p=(1+x+x^2)^n / n!); for (k=1, n, p = deriv(p)); Vecrev(p); \\ Michel Marcus, Jun 22 2019 CROSSREFS Cf. A002426 (first column), A000984 (main diagonal), A122868 (row sums). Sequence in context: A099162 A187327 A271716 * A023832 A080235 A198516 Adjacent sequences:  A220175 A220176 A220177 * A220179 A220180 A220181 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 06 2012 STATUS approved

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Last modified January 23 10:11 EST 2022. Contains 350510 sequences. (Running on oeis4.)