A073157 -id:A073157 - cp's OEIS frontend

A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.

1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, 1840680, 9792986, 52296799, 280163091, 1504969409, 8103433329, 43722788132, 236340999038, 1279602656590, 6938126362948, 37668424608552, 204751452911832, 1114151447523038

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A137634, A137635, A137637, A137638, A073157.

{a(n)=sum(k=0,n,binomial(2*k+1,k)*binomial(2*k+1,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G,n,x)}

G.f.: A(x) = R(x)*G(x), where R(x) = 1/sqrt(1-4x(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4x(1+x)^2))/(2x(1+x)) is the g.f. of A073157.
D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-6*n-1)*a(n-2) +2*(-6*n+1)*a(n-3) +2*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Jun 23 2023
a(n) ~ sqrt((172 + (86*(78905 - 519*sqrt(129)))^(1/3) + (86*(78905 + 519*sqrt(129)))^(1/3))/129) * ((4 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 25 2023

A137637 a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.

Original entry on oeis.org

1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, 3868024, 20785035, 111931154, 603938905, 3264309644, 17671408012, 95800342628, 520022296700, 2826089180652, 15374990077568, 83727902852188, 456370687687082

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A137634, A137635, A137636, A137638, A073157.

{a(n)=sum(k=0,n,binomial(2*k+2,k)*binomial(2*k+2,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G^2,n,x)}

G.f.: A(x) = R(x)*G(x)^2, where R(x) = 1/sqrt(1-4*x*(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4*x*(1+x)^2))/(2*x*(1+x)) is the g.f. of A073157.

A073154 Triangle of numbers relating two sequences (A073157 and A073155).

Original entry on oeis.org

1, 2, 4, 5, 9, 14, 18, 28, 38, 56, 70, 106, 131, 167, 237, 293, 433, 523, 613, 753, 1046, 1283, 1869, 2219, 2543, 2893, 3479, 4762, 5808, 8374, 9839, 11099, 12359, 13824, 16390, 22198

Offset: 0

Paul D. Hanna, Jul 29 2002

			a(4,0)=a(3,3)+a(2,2)=56+14=70.
a(5,2)=A073157(0)*A073157(5)+A073157(1)*A073157(4)+A073157(2)*A073157(3)= 1*293+2*70+5*18=523.
Rows:
  {1};
  {2,4};
  {5,9,14};
  {18,28,38,56};
  {70,106,131,167,237};
  {293,433,523,613,753,1046};
  {1283,1869,2219,2543,2893,3479,4762};
  ...

Cf. A073153, A073155, A073156, A073157, A073152.

Triangle {a(n, k), n >= 0, 0<=k<=n} defined by: a(0, 0)=1, a(n, 0)=A073157(n), a(n, n)=A073155(n+1), a(n, 0)=a(n-1, n-1) + a(n-2, n-2), a(n, k)=sum{j=0..k} A073157(j) * A073157(n-j).

G.f.: Sum_{n>=0, 0<=k<=n} a(n, k) x^n y^k = A(x*y)(A(x) - y A(x*y))/(1 - y) where A(x) = (1 - (1 - 4 x (1 + x)^2)^(1/2))/(2 x (1 + x)) is the o.g.f. for A073157. - David Callan, Aug 16 2006

A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).

Original entry on oeis.org

1, 2, 9, 56, 400, 3095, 25240, 213633, 1859006, 16527544, 149472480, 1370794835, 12718060947, 119158146283, 1125816405458, 10714275588727, 102615375322564, 988302823695146, 9565859385140272, 93000625498797314, 907782305262566776, 8892941663606408172

Offset: 0

Paul D. Hanna, Oct 31 2011

The radius of convergence of g.f. A(x) is r = 0.095007017562450871521918431664620... with A(r) = 1.6228790124092133906198298670423120590101223122... where y=A(r) satisfies 2*y^5 + 6*y^4 - 18*y^3 + 6*y^2 - 3 = 0.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 22*x^2 + 148*x^3 + 1105*x^4 + 8798*x^5 + 73196*x^6 +...
A(x)^3 = 1 + 6*x + 39*x^2 + 284*x^3 + 2223*x^4 + 18267*x^5 + 155445*x^6 +...
A(x)^4 = 1 + 8*x + 60*x^2 + 472*x^3 + 3878*x^4 + 32948*x^5 + 287300*x^6 +...
where A(x) = 1 + x*(A(x) + A(x)^3) + x^2*A(x)^4.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x + (1 + 2^2*A(x)^2 + A(x)^4)*x^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5/5 +...
more explicitly,
log(A(x)) = 2*x + 14*x^2/2 + 122*x^3/3 + 1118*x^4/4 + 10557*x^5/5 + 101642*x^6/6 + 991916*x^7/7 +...

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

Cf. A215623, A215624, A215654, A007863, A036765, A198951, A181734, A073157, A364375.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x*AGF^3) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 19 2013 *)

a(n):=sum((sum((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1),k,0,j))*(-1)^(n-j)*binomial(2*n-j,n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(2*j))*x^m/m))); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2 +x^3*O(x^n)))),n)}

{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*A)*(1 + x*(A+x*O(x^n))^3));polcoeff(A,n)}

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A073157 (Schroeder n-paths containing no FFs).
The formal inverse of g.f. A(x) is (sqrt((1-x^2)^2 + 4*x^3) - (1+x^2)) / (2*x^3).
D-finite with recurrence: 2*n*(n+1)*(2*n+1)*(1275*n^5 - 11696*n^4 + 36827*n^3 - 40618*n^2 - 5828*n + 25368)*a(n) = 6*n*(2*n - 1)*(7650*n^6 - 66351*n^5 + 183953*n^4 - 102147*n^3 - 314787*n^2 + 450754*n - 137760)*a(n-1) - 6*(n-1)*(2*n - 3)*(34425*n^6 - 281367*n^5 + 690471*n^4 - 86579*n^3 - 1831014*n^2 + 2230808*n - 685440)*a(n-2) + 6*(22950*n^8 - 279378*n^7 + 1275447*n^6 - 2461807*n^5 + 518525*n^4 + 5756973*n^3 - 9486182*n^2 + 5962912*n - 1303680)*a(n-3) - 6*(22950*n^8 - 313803*n^7 + 1633059*n^6 - 3736233*n^5 + 1886879*n^4 + 7909228*n^3 - 16107824*n^2 + 11531408*n - 2756544)*a(n-4) + 3*(n-4)*(3*n - 14)*(3*n - 7)*(1275*n^5 - 5321*n^4 + 2793*n^3 + 12437*n^2 - 16992*n + 5328)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 10.5255382776611313... is the root of the equation -27 + 108*d - 108*d^2 + 324*d^3 - 72*d^4 + 4*d^5 = 0 and c = 0.5321376859604656812266678970406658537671... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1)))*(-1)^(n-j)*binomial(2*n-j,n-j)). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1). - Seiichi Manyama, Jul 19 2023

A137635 a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.

Original entry on oeis.org

1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, 826992, 4376408, 23267332, 124179570, 664919780, 3570265000, 19216805476, 103652442922, 560127574340, 3031887311256, 16435458039076, 89213101943000, 484839755040768, 2637805800869740, 14365506336197816

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

Diagonal of rational function 1/(1 - (x + y + x^2*y + x*y^2)). - Gheorghe Coserea, Aug 31 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A137634, A137636, A137637, A137638, A073157.

CoefficientList[Series[1/Sqrt[1 - 4*x*(1 + x)^2],{x,0,50}],x] (* Stefano Spezia, Sep 01 2018 *)
Table[Sum[Binomial[2k,k]Binomial[2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Dec 31 2018 *)
a[n_]:=Binomial[2n, n]HypergeometricPFQ[{(1-2*n)/3, 2(1-n)/3, -2n/3}, {1/2-n, 1/2-n}, -3^3/2^4]; Array[a,24,0] (* Stefano Spezia, Jul 11 2024 *)

a(n)=sum(k=0,n,binomial(2*k,k)*binomial(2*k,n-k));

a(n)=polcoeff(1/sqrt(1-4*x*(1+x +x*O(x^n))^2),n,x);  /* Using the g.f.: */

G.f.: A(x) = 1/sqrt(1 - 4x(1+x)^2).
D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +8*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020
a(n) = binomial(2*n, n)*hypergeom([(1-2*n)/3, 2*(1-n)/3, -2*n/3], [1/2-n, 1/2-n], -3^3/2^4). - Stefano Spezia, Jul 11 2024

A073155 Leftmost column sequence of triangle A073153.

Original entry on oeis.org

1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460

Offset: 0

Paul D. Hanna, Jul 29 2002

			a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

Cf. A073153, A073156, A073157.

Cf. A052709, A360076, A360082, A360083.

Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - Vladeta Jovovic, Oct 10 2003
a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - Paul Barry, Jul 09 2006
Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - Ilya Gutkovskiy, Jul 10 2020

More terms from Vladeta Jovovic, Oct 10 2003

A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

Original entry on oeis.org

1, 2, 7, 39, 242, 1634, 11631, 85957, 653245, 5072862, 40077807, 321106623, 2602911282, 21308131235, 175909559897, 1462846379247, 12242600576066, 103035285071630, 871490142773640, 7404121610615520, 63157400073057627, 540689217572662413, 4644083121177225292

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A073157, A364337, A364338, A364339.

Cf. A198953, A212071, A215623, A215654, A216359.

A364336 := proc(n)
    add( binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364336(n),n=0..80); # R. J. Mathar, Jul 25 2023

nmax = 80; A[_] = 1;
Do[A[x_] = (1 + x)*(1 + x*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

a(n) = sum(k=0, n, binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
D-finite with recurrence -2*n*(2*n+1)*a(n) +(3*n^2+23*n-14)*a(n-1) +(207*n^2 -635*n +494)*a(n-2) +2*(397*n^2 -2031*n +2600)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Sep 10 2024: (Start)
x/series_reversion(x*A(x)) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + ..., the g.f. of A216359.
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + ..., the g.f. of A215623. (End)

A364337 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 9, 68, 580, 5406, 53270, 545844, 5757332, 62094217, 681653493, 7591431752, 85558696024, 974024788280, 11184192097016, 129378232148016, 1506363564912368, 17639001584452320, 207593804132718948, 2454236122156830254, 29132714097692056954, 347086786035103983446

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..907

Cf. A073157, A364336, A364338, A364339.

Cf. A215623, A215715, A234461, A239107.

terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^4) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(4*k+1, n-k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(4*k+1,n-k) / (4*k+1).

A143330 G.f. satisfies: A(x) = (1 + x*A(x)^2)/(1 - x^2).

Original entry on oeis.org

1, 1, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047, 76459765772375

Offset: 0

Paul D. Hanna, Aug 08 2008

Diagonal sums of A060693. - Paul Barry, Feb 11 2009
Starting with the second 1 and inserting a 2 between the 1 and 3: (1, 2, 3, 8, 25, 83, ...) the INVERT transform of that sequence deletes the 2, getting (1, 3, 8, 25, 83, ...). - Gary W. Adamson, Jun 24 2015
Number of Schroeder-like (see A006318) excursions (paths on or above height 0 beginning and ending at height 0) of semilength n, with steps U=(1,1), D=(1,-1), and H=(4,0). - Alexander Burstein, May 21 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, all internal nodes have weight 1, and leaf nodes have weights in {1,2}. - John Tyler Rascoe, Jun 06 2025

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + 1041*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Joseph P. S. Kung and Anna de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012

Cf. A000108, A006318, A060693, A073157.

a := proc(n) option remember; if n <= 3 then return [1, 1, 3, 8][n + 1] fi;
((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) end:
seq(a(n), n = 0..25); # Peter Luschny, Jan 25 2023

CoefficientList[Series[(1 - x^2 - Sqrt[1 - 4 x - 2 x^2 + x^4])/(2 x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=polcoeff((1-x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x),n)}

G.f.: A(x) = (1-x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
From Paul Barry, Feb 11 2009: (Start)
G.f.: 1/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-...))))) (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,k)*A000108(n-2k). (End)
D-finite with recurrence (n+1)*a(n) +(n+2)*a(n-1) +2*(17-11n)*a(n-2) +10*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Dec 11 2011
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.439109106851354261627... is the root of the equation 1 - 2*d^2 - 4*d^3 + d^4 = 0 and c = 1/2*sqrt(d*(d^2+3)/(d^2-1)) = 1.16064231... - Vaclav Kotesovec, Feb 03 2014
G.f. satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*x^k*A(x)^(j-k). - Ilya Gutkovskiy, Apr 11 2019
G.f.: 1/G(x), with G(x) = 1-(x+x^2)/(1-x/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 11 2023
From Peter Luschny, Jan 25 2023: (Start)
a(n) = CatalanNumber(n)*hypergeom([-n/2, -n/2, -n/2 - 1/2, -n/2 + 1/2], [-(2*n)/3, -(2*n)/3 + 1/3, -(2*n)/3 + 2/3], -16/27).
a(n) = ((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) for n >= 4. (End)
G.f. A(x) = -x + (1/x)*series_reversion(x*G(-x)), where G(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ... is the g.f. of A073157. - Peter Bala, Aug 27 2024

Minor edits by Vaclav Kotesovec, Mar 31 2014

A364338 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 11, 105, 1140, 13555, 170637, 2235472, 30161255, 416248640, 5848462880, 83378361111, 1203100853951, 17537182300140, 257858115407535, 3819894878557990, 56958234329850060, 854192593184162160, 12875579347191388830, 194963091634569681550, 2964229359714424159370, 45234864131654311730160

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..827

Cf. A073157, A364336, A364337, A364339.

Cf. A215624, A234525, A239108, A364331, A364335.

terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^5) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(5*k+1, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(5*k+1,n-k) / (5*k+1).

cp's OEIS Frontend

A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A137637 a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A073154 Triangle of numbers relating two sequences (A073157 and A073155).

Views

Author

Keywords

Examples

Crossrefs

Formula

A198953 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

PARI

Formula

A137635 a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A073155 Leftmost column sequence of triangle A073153.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A364337 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^4).

Views

Author

Keywords

Links

Crossrefs

Programs

A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).