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 A025749 4th order Patalan numbers (generalization of Catalan numbers). 8
 1, 1, 6, 56, 616, 7392, 93632, 1230592, 16612992, 228890112, 3204461568, 45445091328, 651379642368, 9419951751168, 137262154088448, 2013178259963904, 29694379334467584, 440175505428578304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014 and J. Int. Seq. 18 (2015) # 15.3.3 FORMULA G.f.: (5-(1-16*x)^(1/4))/4. a(n) = 4^(n-1)*3*A034176(n-1)/n!, n >= 2; 3*A034176(n-1)=(4*n-5)(!^4) := product(4*j-5, j=2..n). - Wolfdieter Lang a(n):=(4^(n-1)*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=0..k, binomial(j,n-3*k+2*j-1)*4^(j-k)*binomial(k,j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1))))/n. [Vladimir Kruchinin, Apr 01 2011] n*a(n) +4*(-4*n+5)*a(n-1)=0. - R. J. Mathar, Apr 05 2018 MATHEMATICA a[n_] := (4^(n-1)*Sum[ Binomial[n+k-1, n-1]*Sum[ Binomial[j, n-3*k+2*j-1] * 4^(j-k) * Binomial[k, j] * 3^(-n+3*k-j+1) * 2^(n-3*k+j-1) * (-1)^(n-3*k+2*j-1), {j, 0, k}], {k, 1, n-1}])/n; a[0] = a[1] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Mar 05 2013, after Vladimir Kruchinin *) PROG (Maxima) a(n):=(4^(n-1)*sum(binomial(n+k-1, n-1)*sum(binomial(j, n-3*k+2*j-1)*4^(j-k)*binomial(k, j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1), j, 0, k), k, 1, n-1))/n; /* Vladimir Kruchinin, Apr 01 2011 */ CROSSREFS Equals 2^n * A048779(n), n>1. Sequence in context: A048348 A227384 A199755 * A297705 A231690 A285166 Adjacent sequences:  A025746 A025747 A025748 * A025750 A025751 A025752 KEYWORD nonn AUTHOR STATUS approved

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Last modified January 21 16:22 EST 2020. Contains 331114 sequences. (Running on oeis4.)