A114496 -id:A114496 - cp's OEIS frontend

A003168 Number of blobs with 2n+1 edges.

1, 1, 4, 21, 126, 818, 5594, 39693, 289510, 2157150, 16348960, 125642146, 976789620, 7668465964, 60708178054, 484093913917, 3884724864390, 31348290348086, 254225828706248, 2070856216759478, 16936016649259364

Offset: 0

N. J. A. Sloane

a(n) is the number of ways to dissect a convex (2n+2)-gon with non-crossing diagonals so that no (2m+1)-gons (m>0) appear. - Len Smiley
a(n) is the number of plane trees with 2n+1 leaves and all non-leaves having an odd number > 1 of children. - Jordan Tirrell, Jun 09 2017
a(n) is the number of noncrossing cacti with n+1 nodes. See A361242. - Andrew Howroyd, Mar 07 2023

			a(2)=4 because we may place exactly one diagonal in 3 ways (forming 2 quadrilaterals), or not place any (leaving 1 hexagon).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Thomas H. Bertschinger, Joseph Slote, Olivia Claire Spencer, and Samuel Vinitsky, The Mathematics of Origami, Undergrad Thesis, Carleton College (2016).
D. Birmajer, J. B. Gil, and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
Frédéric Chapoton and Philippe Nadeau, Combinatorics of the categories of noncrossing partitions, Séminaire Lotharingien de Combinatoire 78B (2017), Article #37.
Michael Drmota, Anna de Mier, and Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See p. 116, B_b(z). - N. J. A. Sloane, May 18 2014
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 415
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Jean-Christophe Novelli and Jean-Yves Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra, arXiv:2106.08257 [math.CO], 2021-2022.
Ronald C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) No. 1, 47-63.
L. Smiley, Even-gon reference
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 30.

Cf. A049124 (no 2m-gons).
Cf. A003169, A100327, A114496, A007318, A361242.
Row sums of A102537, A243662. Column 2 of A336573.

import Data.List (transpose)
a003168 0 = 1
a003168 n = sum (zipWith (*)
   (tail $ a007318_tabl !! n)
   ((transpose $ take (3*n+1) a007318_tabl) !! (2*n+1)))
   `div` fromIntegral n
-- Reinhard Zumkeller, Oct 27 2013

Order := 40; solve(series((A-2*A^3)/(1-A^2),A)=x,A);
A003168 := n -> `if`(n=0,1,A100327(n)/2): seq(A003168(n),n=0..20); # Peter Luschny, Jun 10 2017

a[0] = 1; a[n_] = (2^(-n-1)*(3n)!* Hypergeometric2F1[-1-2n, -2n, -3n, -1])/((2n+1)* n!*(2n)!); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 25 2011, after Vladimir Kruchinin *)

a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^3)/(1-x^2)+O(x^(2*n+2))),2*n+1))

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x*A)/(1-x*A)^2); sum(k=0,n,polcoeff(A^(n-k),k))} \\ Paul D. Hanna, Nov 17 2004

seq(n) = Vec( 1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n)) ) \\ Andrew Howroyd, Mar 07 2023

a(n) = Sum_{k=1..n} binomial(n, k)*binomial(2*n+k, k-1)/n.
G.f.: A(x) = Sum_{n>=0} a(n)*x^(2*n+1) satisfies (A-2*A^3)/(1-A^2)=x. - Len Smiley.
D-finite with recurrence 4*n*(2*n + 1)*(17*n - 22)*a(n) = (1207*n^3 - 2769*n^2 + 1850*n - 360)*a(n - 1) - 2*(17*n - 5)*(n - 2)*(2*n - 3)*a(n - 2). - Vladeta Jovovic, Jul 16 2004
G.f.: A(x) = 1/(1-G003169(x)) where G003169(x) is the g.f. of A003169. - Paul D. Hanna, Nov 17 2004
a(n) = JacobiP(n-1,1,n+1,3)/n for n > 0. - Mark van Hoeij, Jun 02 2010
a(n) = (1/(2*n+1))*Sum_{j=0..n} (-1)^j*2^(n-j)*binomial(2*n+1,j)*binomial(3*n-j,2*n). - Vladimir Kruchinin, Dec 24 2010
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  3, 3, 1
  5, 5, 3, 1
  7, 7, 5, 3, 1
  9, 9, 7, 5, 3, 1
  ... (End)
a(n) ~ sqrt(14+66/sqrt(17)) * (71+17*sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(4*n+4)). - Vaclav Kotesovec, Jul 01 2015
From Peter Bala, Oct 05 2015: (Start)
a(n) = (1/n) * Sum_{i = 0..n} 2^(n-i-1)*binomial(2*n,i)* binomial(n,i+1).
O.g.f. = 1 + series reversion( x/((1 + 2*x)*(1 + x)^2) ).
Logarithmically differentiating the modified g.f. 1 + 4*x + 21*x^2 + 126*x^3 + 818*x^4 + ... gives the o.g.f. for A114496, apart from the initial term. (End)
G.f.: A(x) satisfies A = 1 + x*A^3/(1-x*A^2). - Jordan Tirrell, Jun 09 2017
a(n) = A100327(n)/2 for n>=1. - Peter Luschny, Jun 10 2017

A156894 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n+k-1,k).

Original entry on oeis.org

1, 3, 19, 138, 1059, 8378, 67582, 552576, 4563235, 37972290, 317894394, 2674398268, 22590697614, 191475925332, 1627653567916, 13870754053388, 118464647799075, 1013709715774130, 8689197042438274, 74594573994750972, 641252293546113434, 5519339268476249676, 47558930664216470628

Offset: 0

Paul Barry, Feb 17 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on A156894

Cf. A002003, A003169, A103885, A114496, A123164.

A156894:= func< n | (&+[ Binomial(n,k)*Binomial(2*n+k-1,k): k in [0..n]]) >;
[A156894(n): n in [0..30]]; // G. C. Greubel, Jan 06 2022

a := n -> hypergeom([-n, 2*n], [1], -1);
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Aug 02 2014

Table[Sum[Binomial[n,k]Binomial[2n+k-1,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 12 2014 *)

a(n) = if (n < 1, 1, sum(k=0, n, binomial(n,k)*binomial(2*n+k-1,k)));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

[round( hypergeometric([-n, 2*n], [1], -1) ) for n in (0..30)] # G. C. Greubel, Jan 06 2022

a(n) = [x^n] ((1+x)/(1-x)^2)^n.
a(n) = (4*(n+1)*(2*n+1)*A003169(n+1) - (5*n+1)*(2*n-1)*A003169(n))/(17*n + 5) for n>0. - Mark van Hoeij, Jul 14 2010
a(n) = Hypergeometric2F1([-n, 2*n], [1], -1). - Peter Luschny, Aug 02 2014
Conjecture: 64*n*(2*n-1)*a(n) -16*(89*n^2 -134*n +63)*a(n-1) +4*(661*n^2 -2619*n +2576)*a(n-2) -3*(119*n^2 -713*n +1092)*a(n-3) +6*(2*n-7)*(n-4)*a(n-4) = 0. - R. J. Mathar, Feb 05 2015
Conjecture: 16*n*(782*n +5365)*(2*n-1)*a(n) +8*(3128*n^3 -362053*n^2 +593930*n -290328)*a(n-1) -3*(726869*n^3 -5105981*n^2 +11667946*n -8715544)*a(n-2) +158*(2*n-5)*(n-3)*(391*n -764)*a(n-3) = 0. - R. J. Mathar, Feb 05 2015
Conjecture: 4*n*(2*n-1)*(17*n^2 -52*n +39)*a(n) -(1207*n^4 -4899*n^3 +6692*n^2 -3504*n +576)*a(n-1) +2*(n-2)*(2*n-3)*(17*n^2 -18*n +4)*a(n-2) = 0. - R. J. Mathar, Feb 05 2015 [the Maple command sumrecursion (binomial(n,k) * binomial(2*n+k-1,k), k, a(n)) verifies this recurrence. - Peter Bala, Oct 05 2015 ]
a(n) ~ sqrt(578 + 306*sqrt(17)) * (71 + 17*sqrt(17))^n / (17 * sqrt(Pi*n) * 2^(4*n+2)). - Vaclav Kotesovec, Feb 05 2015
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 14*x^2 + 79*x^3 + ... is the o.g.f. of A003169 (taken with offset 0). - Peter Bala, Oct 05 2015
From Peter Bala, Mar 20 2020: (Start)
a(p) == 3 ( mod p^3 ) for prime p >= 5. Cf. A002003, A103885 and A119259.
More generally, we conjecture that a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. (End)

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

Original entry on oeis.org

1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483

Offset: 0

Yifan Zhang, Dec 13 2020

nonn

Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.

Seiichi Manyama, Table of n, a(n) for n = 0..838
F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy

Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).

Cf. A026000, A114496.

Table[Sum[Binomial[n,k]*Binomial[2n+k,k]*2^k,{k,0,n}],{n,0,20}] (* or *)
Table[Hypergeometric2F1[-n,1+2 n,1,-2],{n,0,20}] (* Stefano Spezia, Dec 17 2020 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021

a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
From Vaclav Kotesovec, May 11 2021: (Start)
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)

More terms from Stefano Spezia, Dec 17 2020

A156886 a(n) = Sum_{k=0..n} C(n,k)C(3n+k,k).

Original entry on oeis.org

1, 5, 43, 416, 4239, 44485, 475780, 5156548, 56437231, 622361423, 6904185523, 76964141600, 861408728964, 9673849095708, 108954068684616, 1230185577016156, 13920106205444335, 157814104889538739

Offset: 0

Paul Barry, Feb 17 2009

a(n)=[x^n](1+5x+9x^2+7x^3+2x^4)^n. The coefficients (1,5,9,7,2) are the 5th row of A029635.

Robert Israel, Table of n, a(n) for n = 0..937
P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Cf. A001850, A114496, A029635, A156887.

A156886 := proc(n)
    add(binomial(n,k)*binomial(3*n+k,k), k = 0..n);
end proc:
seq(A156886(n), n = 0..20); # Peter Bala, Feb 11 2018

a[n_] := Sum[ Binomial[n, k] Binomial[3n + k, k], {k, 0, n}]; Array[a, 21, 0] (* Robert G. Wilson v, Feb 11 2018 *)

From Peter Bala, Feb 11 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(3*n+k,n)*2^k.
a(n) = Sum_{k = 0..n} C(n,k)*C(3*n,k)*2^(n-k),
12*n*(3*n-1)*(3*n-2)*(238*n^2 - 663*n + 457)*a(n) = 2*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - (3*n-3)*(3*n-4)*(3*n-5)*(238*n^2 - 187*n + 32)*a(n-2). (End)
a(n) = P_n(0,2*n,3) where P_n(a,b,x) is the n-th Jacobi polynomial with parameters a and b. - Robert Israel, Feb 11 2018
a(n) ~ sqrt(1/3 + 11/(12*sqrt(7))) * ((316 + 119*sqrt(7))/54)^n / sqrt(Pi*n). - Vaclav Kotesovec, Jan 09 2023

A156887 a(n) = Sum_{k=0..n} C(n,k)C(4n+k,k).

Original entry on oeis.org

1, 6, 64, 768, 9708, 126386, 1676956, 22548168, 306167324, 4188703512, 57649462164, 797294161824, 11071026740964, 154250752864812, 2155368246401224, 30192512693210888, 423859798484668188, 5961793387214958792, 83998039356129372448, 1185277027372535468544

Offset: 0

Paul Barry, Feb 17 2009

a(n)=[x^n] (1+6x+14x^2+16x^3+9x^4+2x^5)^n. The coefficients (1,6,14,16,9,2) are the 6th row of A029635.

Seiichi Manyama, Table of n, a(n) for n = 0..862
P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Cf. A001850, A114496, A156886, A359646.

A156887 := proc(n)
    add(binomial(n,k)*binomial(4*n+k,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 25 2015

Table[Sum[Binomial[n,k]Binomial[4n+k,k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 24 2018 *)

{a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+k, k))} \\ Seiichi Manyama, Feb 02 2019

Conjecture: 576*n*(32901928701*n-65877527665)*(4*n-3)*(2*n-1)*(4*n-1)*a(n) +(-8795436181229177*n^5 +35251410418024655*n^4 -47934714902592853*n^3 +29414167990853161*n^2 -9060238526902314*n +1466702211905280)*a(n-1) +8*(10299715469615*n^5 -136961193094719*n^4 +872530072905392*n^3 -2699499511785411*n^2 +3902106377543903*n -2123717948975100)*a(n-2) -64*(2*n-5)*(4*n-9)*(n-2)*(27741827*n-2925269736)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Feb 25 2015
From Peter Bala, Feb 11 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(4*n+k,n)*2^k.
a(n) = Sum_{k = 0..n} C(n,k)*C(4*n,k)*2^(n-k).
8*(4*n)*(4*n-1)*(4*n-2)**(4*n-3)*(4633*n^3-19662*n^2+27593*n-12804)*a(n) = (137604733*n^7-859190528*n^6+2179882848*n^5-2890753162*n^4+2144669963*n^3-880916550*n^2+182941416*n-14515200)*a(n-1) - (4*n-4)*(4*n-5)*(4*n-6)*(4*n-7)*(4633*n^3-5763*n^2+2168*n-240)*a(n-2). A proof of Mathar's conjectured third-order recurrence above follows easily using this second-order recurrence.  (End)
a(n) ~ sqrt(5 + 33/sqrt(41)) * ((29701 + 4633*sqrt(41)))^n / (sqrt(Pi*n) * 2^(12*n + 2)). - Vaclav Kotesovec, Jan 09 2023

A114497 Numerators of Apéry-style convergents to 4/11 log 2.

Original entry on oeis.org

0, 1, 865, 12643, 13619843, 323746091, 115021083581, 2224431220019, 161734891776923, 57221149255770431, 7283680944060726343, 655528909167704911, 8662089991175424531107, 33378033361711480198829

Offset: 0

Eric Rowland, Dec 01 2005

Denominators are given by A114498.

Cf. A114496, A114498.

A114498 Denominators of Apéry-style convergents to 4/11 log 2.

Original entry on oeis.org

1, 4, 3432, 50160, 54035520, 1284433920, 456335953920, 8825233697280, 641668847339520, 227019837729484800, 28897358537888096256, 2600752842642898944, 34366074253507451879424, 132424388815310703820800

Offset: 0

Eric Rowland, Dec 01 2005

Numerators are given by A114497.

Cf. A114496, A114497.

A359646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,k).

Original entry on oeis.org

1, 7, 89, 1273, 19181, 297662, 4707971, 75459496, 1221388525, 19919031781, 326797222834, 5387618403526, 89178832899887, 1481143718244912, 24671054686539336, 411966653603163008, 6894167059382069485, 115593504497163747167, 1941434442814233362939, 32656575110841643234631

Offset: 0

Vaclav Kotesovec, Jan 09 2023

nonn

In general, for m>0, Sum_{k=0..n} binomial(n,k) * binomial(m*n+k,k) ~ (m+c) / sqrt(2*Pi*c*m * (m*(2-c)+c)*n) * d^n, where d = (m+c)^(m+c) / ((1-c)^(1-c) * c^(2*c) * m^m) and c = (sqrt(m^2 + 6*m + 1) + 1 - m)/4.

Equivalently, d = (3 + m + sqrt(1 + m*(6 + m))) * (1 + 3*m + sqrt(1 + m*(6 + m)))^m / (2^(2*m + 1) * m^m).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001850, A114496, A156886, A156887.

Table[Sum[Binomial[n, k]*Binomial[5*n+k, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(5*n+k,k)) \\ Andrew Howroyd, Jan 09 2023

a(n) ~ sqrt(3/10 + 23/(20*sqrt(14))) * ((108007 + 28854*sqrt(14))/12500)^n / sqrt(Pi*n).

A268543 The diagonal of 1/(1 - (y + z + x z + x w + x y w)).

Original entry on oeis.org

1, 8, 156, 3800, 102340, 2919168, 86427264, 2626557648, 81380484900, 2559296511200, 81443222791216, 2616761264496288, 84749038859067856, 2763262653898544000, 90615128199047200800, 2986287891921565639200, 98841887070519004625700

Offset: 0

N. J. A. Sloane, Feb 29 2016

From Gheorghe Coserea, Jul 03 2016: (Start)
Also diagonal of rational function R(x,y,z) = 1/(1 - x - y - z - x*y).
Annihilating differential operator: x*(2*x+3)*(16*x^2-71*x+2)*Dx^2 + 2*(32*x^3+x^2-213*x+3)*Dx + 8*x^2+48*x-48.
(End)

Gheorghe Coserea, Table of n, a(n) for n = 0..310
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A268545-A268555, A000984, A114496.

A268543 := proc(n)
    1/(1-y-z-x*z-x*w-x*y*w) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,n) ;
    coeftayl(%,z=0,n) ;
    coeftayl(%,w=0,n) ;
end proc:
seq(A268543(n),n=0..40) ; # R. J. Mathar, Mar 11 2016
#alternative program
with(combinat):
seq(binomial(2*n,n)*add(binomial(n,k)*binomial(2*n+k,k), k = 0..n), n = 0..20); # Peter Bala, Jan 27 2018

CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12}, {1}, 1728*x^3*(2 - 71*x + 16*x^2)/(1 - 32*x + 16*x^2)^3]*(1 - 32*x + 16*x^2)^(-1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

my(x='x, y='y, z='z, w='w);
R = 1/(1 - x - y - z - x*y);
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(10, R, [x,y,z])

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom_sym([1/12,5/12],[1],1728*x^3*(16*x^2-71*x+2)/(16*x^2-32*x+1)^3, N)/(16*x^2-32*x+1)^(1/4))  \\ Gheorghe Coserea, Jul 03 2016

Conjecture: 2*n^2*(17*n-23)*a(n) +(-1207*n^3+2840*n^2-1897*n+360)*a(n-1) + 4*(17*n-6)*(-3+2*n)^2*a(n-2) = 0. - R. J. Mathar, Mar 11 2016
G.f.: hypergeom([1/12, 5/12], [1], 1728*x^3*(2-71*x+16*x^2)/(1-32*x+16*x^2)^3)*(1-32*x+16*x^2)^(-1/4). - Gheorghe Coserea, Jul 01 2016
0 = x*(2*x+3)*(16*x^2-71*x+2)*y'' + 2*(32*x^3+x^2-213*x+3)*y' + (8*x^2+48*x-48)*y, where y is the g.f. - Gheorghe Coserea, Jul 03 2016
a(n) ~ sqrt(3 + 13/sqrt(17)) * (71+17*sqrt(17))^n / (Pi * n * 2^(2*n + 3/2)). - Vaclav Kotesovec, Jul 05 2016
From Peter Bala, Jan 27 2018: (Start)
a(n) = binomial(2*n,n)*Sum_{k = 0..n} binomial(n,k)* binomial(2*n+k,k) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0]}). Using this binomial sum, Maple confirms the above recurrence of Mathar.
a(n) = A000984(n)*A114496(n). (End)

A306280 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n^2+k,k).

Original entry on oeis.org

1, 3, 26, 416, 9708, 297662, 11306572, 512307336, 26968496504, 1617489748394, 108885682104744, 8129721925098468, 666736347200187804, 59582961423951290184, 5762936296492591067968, 599807329803134064385488, 66843498592187788579795440

Offset: 0

Seiichi Manyama, Feb 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..337

Cf. A001850, A114496, A135860, A156886, A156887.

a[n_] := Sum[Binomial[n,k] * Binomial[n^2+k,k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2019 *)

{a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2+k, k))}

From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ exp(1) * A135860(n).
a(n) ~ exp(n + 3/2) * n^(n - 1/2) / sqrt(2*Pi). (End)

cp's OEIS Frontend

A003168 Number of blobs with 2n+1 edges.

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A156894 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n+k-1,k).

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A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.

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A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k).

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A156887 a(n) = Sum_{k=0..n} C(n,k)*C(4*n+k,k).

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A114497 Numerators of Apéry-style convergents to 4/11 log 2.

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A114498 Denominators of Apéry-style convergents to 4/11 log 2.

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A359646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,k).

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A156894 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n+k-1,k).

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

A156886 a(n) = Sum_{k=0..n} C(n,k)C(3n+k,k).

A156887 a(n) = Sum_{k=0..n} C(n,k)C(4n+k,k).