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 A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows. 1
 1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Note that the following g.f. does NOT yield an integer triangle: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^k * y^k ). LINKS EXAMPLE This triangle begins: 1; 1, 1; 1, 3, 1; 1, 6, 12, 1; 1, 10, 71, 76, 1; 1, 15, 281, 2153, 701, 1; 1, 21, 861, 29166, 129509, 8477, 1; 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1; 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1; 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ... ... G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +... The logarithm of the g.f. equals the series: log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2 + (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3 + (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4 + (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5 + (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +... in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1). PROG (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*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. A184730 (row sums), A181070 (antidiagonal sums), A060946 (diagonal). Cf. related triangles: A219207, A209424, A228904. Sequence in context: A235116 A235114 A272866 * A255918 A102479 A228902 Adjacent sequences:  A228896 A228897 A228898 * A228900 A228901 A228902 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Sep 07 2013 STATUS approved

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Last modified October 16 06:30 EDT 2019. Contains 328050 sequences. (Running on oeis4.)