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 A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows. 8
 1, 1, 1, 1, 4, 1, 1, 32, 32, 1, 1, 487, 3282, 487, 1, 1, 11113, 657573, 657573, 11113, 1, 1, 335745, 209282906, 1513844855, 209282906, 335745, 1, 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1, 1, 565877928, 61162554558200 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Paul D. Hanna, Rows n = 0..30, flattened. FORMULA Column 1 equals A209197. Row sums equal A167006. Antidiagonal sums equal A206830. EXAMPLE This triangle begins: 1; 1, 1; 1, 4, 1; 1, 32, 32, 1; 1, 487, 3282, 487, 1; 1, 11113, 657573, 657573, 11113, 1; 1, 335745, 209282906, 1513844855, 209282906, 335745, 1; 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1; 1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ... G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +... The logarithm of the g.f. equals the series: log(A(x,y)) = (1 + y)*x + (1 + 6*y + y^2)*x^2/2 + (1 + 84*y + 84*y^2 + y^3)*x^3/3 + (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4 + (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +... in which the coefficients form A209330(n,k) = binomial(n^2, n*k). PROG (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, m*j)*y^j))+x*O(x^n)), n, x), k, y)} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A209197, A167006, A206830, A209330 (log), A155200. Sequence in context: A173918 A174412 A177939 * A158390 A228836 A176419 Adjacent sequences:  A209193 A209194 A209195 * A209197 A209198 A209199 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Mar 05 2012 STATUS approved

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Last modified September 22 03:21 EDT 2020. Contains 337289 sequences. (Running on oeis4.)