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 A107111 Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals. 5
 1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 13, 22, 14, 1, 5, 23, 67, 90, 42, 1, 6, 36, 156, 381, 394, 132, 1, 7, 52, 305, 1162, 2307, 1806, 429, 1, 8, 71, 530, 2833, 9192, 14589, 8558, 1430, 1, 9, 93, 847, 5919, 27916, 75819, 95235, 41586, 4862, 1, 10, 118, 1272, 11070, 70098, 286632, 644908, 636925, 206098, 16796 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS First row is the Catalan numbers A100108, second row is the large Schroeder numbers A006318, third row is A062992, fourth row is A007297. As a number triangle, this is T(n,k)=if(k<=n,sum{j=0..k, binomial((n-k)(k+1),k-j)*binomial(k+j,j)}/(k+1),0) with row sums A107112 and diagonal sums A107113. LINKS FORMULA T(n, k)=sum{j=0..k, binomial(n(k+1), k-j)*binomial(k+j, j)}/(k+1) EXAMPLE Array begins 1,1,2,5,14,42,132,... 1,2,6,22,90,394,1806,... 1,3,13,67,381,2307,14589,... 1,4,23,156,1162,9192,75819,... MAPLE A107111 := proc(n, k)     add(binomial(n*(k+1), k-j)*binomial(k+j, j), j=0..k);     %/(k+1) ; end proc: # R. J. Mathar, Aug 02 2016 MATHEMATICA T[n_, k_] := Sum[Binomial[n (k + 1), k - j] Binomial[k + j, j], {j, 0, k}]/(k + 1); Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2020 *) CROSSREFS Sequence in context: A056860 A158825 A247507 * A082037 A163649 A110858 Adjacent sequences:  A107108 A107109 A107110 * A107112 A107113 A107114 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, May 12 2005 STATUS approved

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Last modified January 23 18:30 EST 2021. Contains 340386 sequences. (Running on oeis4.)