A025759 -id:A025759 - cp's OEIS frontend

A025756 3rd-order Vatalan numbers (generalization of Catalan numbers).

1, 1, 4, 22, 139, 949, 6808, 50548, 384916, 2988418, 23559826, 188061592, 1516680130, 12337999870, 101111413540, 833914857316, 6916004156083, 57638242134229, 482444724374734, 4053815358183454, 34181335453533439

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014 and J. Int. Seq. 18 (2015) # 15.3.3

Row sums of triangle A048966, n > 0.

Cf. A000108, A025757, A025758, A025759, A025760, A025761, A025762, A025763.

A025756 := proc(n)
    coeftayl( 3/(2+(1-9*x)^(1/3)), x=0, n);
end proc:
seq(A025756(n), n=0..30); # Wesley Ivan Hurt, Aug 02 2014

Table[SeriesCoefficient[3/(2+(1-9*x)^(1/3)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

a[0]:1$ a[n]:=(1/n)*((9*n-6)*a[n-1]-2*sum(a[k]*a[n-1-k], k, 0, n-1))$ makelist(a[n],n,0,1000); /* Tani Akinari, Aug 02 2014 */

G.f.: 3 / (2+(1-9*x)^(1/3)).
a(n) = Sum_{m=1..n-1} (m/n) * Sum_{k=1..n-m} binomial(k,n-m-k) * 3^k * (-1)^(n-m-k) * binomial(n+k-1,n-1) + 1. - Vladimir Kruchinin, Feb 08 2011
Conjecture: n*(n-1)*a(n) -(n-1)*(19*n-36)*a(n-1) +9*(11*n^2-51*n+60)*a(n-2) -9*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 9^n/(4*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Oct 08 2012
a(n) = (-1)^(n+1) * 3^(2*n+1) * Sum_{k>=0} (-1/2)^(k+1) * binomial(k/3,n). - Seiichi Manyama, Aug 04 2024

A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 7, 69, 783, 9597, 123495, 1643397, 22413183, 311466829, 4392857431, 62702224213, 903886452975, 13138698859677, 192337495360071, 2832859169364261, 41946319269028191, 624009420903043821

Offset: 0

Olivier Gérard

nonn

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.
T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014
T. M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.

a(n), n >= 1, = row sums of triangle A049213.

Cf. A000108, A025756, A025758, A025759, A025760, A025761, A025762, A025763.

Table[SeriesCoefficient[4/(3 + (1 - 16*x)^(1/4)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

G.f.: 4 / (3+(1-16*x)^(1/4)).
a(n) = Sum_{m=1..n-1} (m/n*4^(n-m)) * Sum_{k=1..n-m} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j) + 1. - Vladimir Kruchinin, Feb 08 2011
Conjecture: 5*n*(n-1)*(n-2)*a(n) -(239*n-600)*(n-1)*(n-2)*a(n-1) +24*(n-2)*(158*n^2-953*n+1445)*a(n-2) +16*(-1232*n^3+13056*n^2-45949*n+53730)*a(n-3) -128*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jul 28 2014
a(n) = (-1)^(n+1) * 4^(2*n+1) * Sum_{k>=0} (-1/3)^(k+1) * binomial(k/4,n). - Seiichi Manyama, Aug 04 2024

A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 11, 171, 3056, 58916, 1191376, 24896436, 532911346, 11617952106, 256966100966, 5750337968926, 129926216608236, 2959472057112396, 67877180959091156, 1566072624078270516, 36319953436423545851

Offset: 0

Olivier Gérard

nonn

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.

a(n), n >= 1, = row sums of triangle A049223.

Cf. A000108, A025756, A025757, A025759, A025760, A025761, A025762, A025763.

# Based on Tani Akinari's formula.
h := (n,j) -> ((-1)^n/(-4)^j)*binomial(j/5,n+1)*hypergeom([1,n+1-j/5],[n+2], 1025): a := n -> 2^8*5^(2*n+1)*add(h(n,j), j=1..4):
seq(round(evalf(a(n),64)),n=0..16); # Peter Luschny, Sep 21 2015

Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

a(n):=(256/205)*41^(-n)*sum(sum((-4)^(-k)*(-1025)^m*binomial(k/5,m),k,0,4),m,0,n);  /* Tani Akinari, Sep 16 2015  */

G.f.: 5 / (4+(1-25*x)^(1/5)).
a(n) = sum(m=1..n-1, 5^(n-m)*m/n * sum(k=1..n-m, binomial(n+k-1,n-1) * sum(i=0..k, binomial(k,i) * 2^(k-i) * sum(j=0..i, binomial(j,-3*i+n-m-k+2*j) * (-1)^(j-i)*5^(j-i)*(-2)^(3*i-n+m+k-j) * binomial(i,j)))))+1. - Vladimir Kruchinin, Feb 09 2011
Conjecture: 41*n*(n-1)*(n-2)*(n-3)*a(n) -3*(1367*n-4100)*(n-1)*(n-2)*(n-3)*a(n-1) +50*(n-2)*(n-3)*(3077*n^2-21533*n+38136)*a(n-2) -250*(n-3)*(10265*n^3-123135*n^2+494446*n-664572)*a(n-3) +1875*(8575*n^4-154250*n^3+1039765*n^2-3112730*n+3491808)*a(n-4) -625*(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-5)=0. - R. J. Mathar, Jul 28 2014
a(n) = (-1)^(n+1) * 5^(2*n+1) * Sum_{k>=0} (-1/4)^(k+1) * binomial(k/5,n). - Seiichi Manyama, Aug 04 2024

A025760 7th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 22, 680, 24074, 917414, 36618492, 1508943612, 63643109727, 2732349490669, 118957846271104, 5237911268468572, 232794783971436296, 10427673857731312064, 470213556090357498728, 21325335129901497816528

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A025756, A025757, A025758, A025759, A025761, A025762, A025763.

Table[SeriesCoefficient[7/(6 + (1 - 49*x)^(1/7)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

G.f.: 7/(6 + (1 - 49*x)^(1/7)).

a(n) = (-1)^(n+1) * 7^(2*n+1) * Sum_{k>=0} (-1/6)^(k+1) * binomial(k/7,n). - Seiichi Manyama, Aug 04 2024

A025761 8th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 29, 1177, 54629, 2726977, 142504685, 7685245225, 424109499317, 23818681210961, 1356315674712509, 78100982458201017, 4538960021319997189, 265837773438037013857, 15672475449746510425485

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A025756, A025757, A025758, A025759, A025760, A025762, A025763.

Table[SeriesCoefficient[8/(7 + (1 - 64 x)^(1/8)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
CoefficientList[Series[8/(7+Surd[1-64x,8]),{x,0,20}],x] (* Harvey P. Dale, Jun 19 2022 *)

G.f.: 8/(7 + (1 - 64*x)^(1/8)).

a(n) = (-1)^(n+1) * 8^(2*n+1) * Sum_{k>=0} (-1/7)^(k+1) * binomial(k/8,n). - Seiichi Manyama, Aug 04 2024

A025762 9th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 37, 1909, 112483, 7123123, 472012183, 32269160215, 2256940619488, 160620490138312, 11588554266307408, 845405636848547320, 62239772654736987376, 4617428134374127211320, 344799253687326693132448

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A025756, A025757, A025758, A025759, A025760, A025761, A025763.

Table[SeriesCoefficient[9/(8 + (1 - 81 x)^(1/9)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

G.f.: 9/(8 + (1 - 81*x)^(1/9)).

a(n) = (-1)^(n+1) * 9^(2*n+1) * Sum_{k>=0} (-1/8)^(k+1) * binomial(k/9,n). - Seiichi Manyama, Aug 04 2024

A025763 10th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 46, 2941, 214486, 16801306, 1376657776, 116346220021, 10057692309166, 884572748725086, 78862377561315156, 7108473985655908626, 646575307673212875996, 59260508917444358016516

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A025756, A025757, A025758, A025759, A025760, A025761, A025762.

Table[SeriesCoefficient[10/(9 + (1 - 100 x)^(1/10)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
CoefficientList[Series[10/(9+Surd[1-100x,10]),{x,0,20}],x] (* Harvey P. Dale, Feb 12 2017 *)

G.f.: 10/(9 + (1 - 100*x)^(1/10)).

a(n) = (-1)^(n+1) * 10^(2*n+1) * Sum_{k>=0} (-1/9)^(k+1) * binomial(k/10,n). - Seiichi Manyama, Aug 04 2024

A049224 A convolution triangle of numbers obtained from A025751.

Original entry on oeis.org

1, 15, 1, 330, 30, 1, 8415, 885, 45, 1, 232254, 26730, 1665, 60, 1, 6735366, 825858, 58320, 2670, 75, 1, 202060980, 25992252, 2003562, 106560, 3900, 90, 1, 6213375135, 830282805, 68351283, 4038741, 174825, 5355, 105, 1, 194685754230

Offset: 1

Wolfdieter Lang

a(n,1) = A025751(n); a(n,1)= 6^(n-1)*5*A034787(n-1)/n!, n >= 2.

G.f. for m-th column: ((1-(1-36*x)^(1/6))/6)^m.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A048966, A049223. Row sums = A025759.

T(n,m):=(m*sum(binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1),k,0,n-i),i,m,n))/n; /* Vladimir Kruchinin, Dec 21 2011 */

a(n, m) = 6*(6*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f.: [(1-(1-36*x)^(1/6))/6]^m=sum(n>=m, T(n,m)*x^n), T(n,m)=(m*sum(i=m..n, binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(k=0..n-i, binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1))))/n. - Vladimir Kruchinin, Dec 21 2011

Showing 1-8 of 8 results.

cp's OEIS Frontend

A025756 3rd-order Vatalan numbers (generalization of Catalan numbers).

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A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

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A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

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A025760 7th-order Vatalan numbers (generalization of Catalan numbers).

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A025761 8th-order Vatalan numbers (generalization of Catalan numbers).

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A025762 9th-order Vatalan numbers (generalization of Catalan numbers).

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A025763 10th-order Vatalan numbers (generalization of Catalan numbers).

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A049224 A convolution triangle of numbers obtained from A025751.

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