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A025750 5th-order Patalan numbers (generalization of Catalan numbers). 5
1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500 (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 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.3.3 .

FORMULA

G.f.: (6-(1-25*x)^(1/5))/5.

a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2; 4*A034301(n-1)= (5*n-6)(!^5) := product(5*j-6, j=2..n). - Wolfdieter Lang

a(n) = (sum(k=0..n-1, (-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(j=0..k, 2^j*binomial(k,j)*sum(i=j..n-k+j-1, binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)))))/n, n>0, a(0)=1. - Vladimir Kruchinin, Dec 10 2011

a(n) = ((-5)^(n-1)*sum(k=1..n, (5)^(n-k)*stirling1(n,k)))/n!, n>0, a(0)=1. - Vladimir Kruchinin, Mar 19 2013

MATHEMATICA

CoefficientList[Series[(6-(1-25x)^(1/5))/5, {x, 0, 20}], x] (* Harvey P. Dale, Dec 06 2012 *)

a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)

PROG

(Maxima)

a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1, n-1)*sum(2^j*binomial(k, j)*sum(binomial(j, i-j)*binomial(k-j, n-3*(k-j)-i-1)*5^(3*(k-j)+i), i, j, n-k+j-1), j, 0, k), k, 0, n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */

(Maxima)

a(n):=if n=0 then 1 else -binomial(1/5, n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */

CROSSREFS

Cf. A034687, A049393.

Sequence in context: A116156 A224124 A212472 * A034325 A335800 A239620

Adjacent sequences:  A025747 A025748 A025749 * A025751 A025752 A025753

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)