A025756 -id:A025756 - cp's OEIS frontend

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

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

A025759 6th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 16, 361, 9346, 260710, 7622290, 230167345, 7116228250, 224012186830, 7152402830440, 230999414308090, 7531444277855740, 247510726140787240, 8189274963276187990, 272537576338530727585, 9116110475685684958810, 306286229879232067776310

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.

a(n), n >= 1, = row sums of triangle A049224.

Cf. A000108, A025756, A025757, A025758, A025760, A025761, A025762, A025763.

Table[SeriesCoefficient[6/(5 + (1 - 36*x)^(1/6)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

a[0]:1$ a[1]:1$ a[2]:16$ a[3]:361$ a[4]:9346$ a[5]:260710$
a[n]:=((n-1)*(n-2)*(n-3)*(n-4)*(78119*n-273420)*a[n-1]-90*(n-2)*(n-3)*(n-4)*(62494*n^2-499959*n+1014685)*a[n-2]+180*(n-3)*(n-4)*(1124856*n^3-15185808*n^2+68852647*n-104826890)*a[n-3]-3240*(n-4)*(1124784*n^4-22496184*n^3+169193274*n^2-567111339*n+714764687)*a[n-4]+5184*(5060556*n^5-139170960*n^4+1530231885*n^3-8408803050*n^2+23092951859*n-25356134300)*a[n-5]+93312*(2*n-11)*(3*n-16)*(3*n-17)*(6*n-31)*(6*n-35)*a[n-6])/(434*n*(n-1)*(n-2)*(n-3)*(n-4));
makelist(a[n],n,0,500);  /* Tani Akinari, Sep 15 2015  */

default(seriesprecision, 40); Vec(6/(5+(1-36*x)^(1/6)) + O(x^30)) \\ Michel Marcus, Sep 15 2015

G.f.: 6/(5+(1-36*x)^(1/6)).
Recurrence: for n>5,
a(n)=((n-1)*(n-2)*(n-3)*(n-4)*(78119*n-273420)*a(n-1)-90*(n-2)*(n-3)*(n-4)*(62494*n^2-499959*n+1014685)*a(n-2)+180*(n-3)*(n-4)*(1124856*n^3-15185808*n^2+68852647*n-104826890)*a(n-3)-3240*(n-4)*(1124784*n^4-22496184*n^3+169193274*n^2-567111339*n+714764687)*a(n-4)+5184*(5060556*n^5-139170960*n^4+1530231885*n^3-8408803050*n^2+23092951859*n-25356134300)*a(n-5)+93312*(2*n-11)*(3*n-16)*(3*n-17)*(6*n-31)*(6*n-35)*a(n-6))/(434*n*(n-1)*(n-2)*(n-3)*(n-4)). - Tani Akinari, Sep 15 2015
a(n) ~ 36^n / (25 * Gamma(5/6) * n^(7/6)) * (1 - 2^(1/3)*sqrt(3)*Gamma(2/3) / (5*sqrt(Pi)*n^(1/6))). - Vaclav Kotesovec, Sep 22 2015
a(n) = (-1)^(n+1) * 6^(2*n+1) * Sum_{k>=0} (-1/5)^(k+1) * binomial(k/6,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

A048966 A convolution triangle of numbers obtained from A025748.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 90, 39, 9, 1, 594, 270, 72, 12, 1, 4158, 1953, 567, 114, 15, 1, 30294, 14580, 4482, 1008, 165, 18, 1, 227205, 111456, 35721, 8667, 1620, 225, 21, 1, 1741905, 867834, 287199, 73656, 15075, 2430, 294, 24, 1, 13586859, 6857136, 2328183, 623106, 136323, 24354, 3465, 372, 27, 1

Offset: 1

Wolfdieter Lang

A generalization of the Catalan triangle A033184.

			Triangle begins:
     1;
     3,    1;
    15,    6,    1;
    90,   39,    9,    1;
   594,  270,   72,   12,    1;
  4158, 1953,  567,  114,   15,    1;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A034000, A049213, A049223, A049224. a(n, 1)= A025748(n), a(n, 1)= 3^(n-1)*2*A034000(n-1)/n!, n >= 2. Row sums = A025756.

a048966 n k = a048966_tabl !! (n-1) !! (k-1)
a048966_row n = a048966_tabl !! (n-1)
a048966_tabl = [1] : f 2 [1] where
   f x xs = ys : f (x + 1) ys where
     ys = map (flip div x) $ zipWith (+)
          (map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))
          (zipWith (*) [1..] ([0] ++ xs))
-- Reinhard Zumkeller, Feb 19 2014

a[n_, m_] /; n >= m >= 1 := a[n, m] = 3*(3*(n-1) - m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m := 0; a[n_, 0] = 0; a[1, 1] = 1; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 26 2011, after given formula *)

a(n, m) = 3*(3*(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. for m-th column: ((1-(1-9*x)^(1/3))/3)^m.

a(n,m) = m/n * sum(k=0..n-m, binomial(k,n-m-k) * 3^k*(-1)^(n-m-k) * binomial(n+k-1,n-1)). - Vladimir Kruchinin, Feb 08 2011

Showing 1-8 of 8 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A025759 6th-order Vatalan numbers (generalization of Catalan numbers).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A048966 A convolution triangle of numbers obtained from A025748.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula