A257142 - OEIS

%I #21 Apr 18 2015 01:07:36

%S 1,3,1,8,7,1,21,34,12,1,55,141,89,18,1,144,534,522,186,25,1,377,1905,

%T 2651,1445,340,33,1,987,6512,12198,9370,3350,568,42,1,2584,21557,

%U 52211,53533,26804,6881,889,52,1

%N Triangle read by rows: T(n,m) = Sum_{k=0..n} C(k-1,m-1)*C(k,m-1)*C(n+k-1,n-k)/m.

%C First column is A001906.

%F G.f.: N(x/(1-x)^2,y), where N(x,y) is the g.f. of Narayana's triangle A001263.

%e 1;

%e 3,1;

%e 8,7,1;

%e 21,34,12,1;

%e 55,141,89,18,1;

%e 144,534,522,186,25,1;

%t Table[Sum[Binomial[k - 1, m - 1] * Binomial[k, m - 1] * Binomial[n + k - 1, n - k]/m, {k, 0, n}], {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Apr 17 2015 *)

%o (Maxima) T(n,m):=sum(binomial(k-1,m-1)*binomial(k,m-1)*binomial(n+k-1,n-k)/m,k,0,n);

%o (PARI) tabl(nn) = {default(seriesprecision, nn+1); pol = subst((1-xx*(1+y)-sqrt((1-xx*(1+y))^2-4*y*xx^2))/(2*xx), xx, x/(1-x)^2) + O(x^nn); for (n=1, nn-1, poly = polcoeff(pol, n, x); for (k=1, n, print1(polcoeff(poly, k, y), ", ");); print(););} \\ _Michel Marcus_, Apr 17 2015

%Y Cf. A001263, A001906.

%K nonn,tabl

%O 1,2

%A _Vladimir Kruchinin_, Apr 16 2015