A032443 -id:A032443 - cp's OEIS frontend

A240879 Self-convolution of Sum(binomial(2*n, i), i=0..n).

1, 6, 31, 150, 699, 3178, 14198, 62604, 273235, 1182786, 5085666, 21743956, 92522206, 392066340, 1655432524, 6967724312, 29245179267, 122442487474, 511487386730, 2132341655556, 8873167793578, 36861311739308, 152895342950196, 633290273209000, 2619653638855214, 10823294835350388

Offset: 0

Fung Lam, Apr 13 2014

nonn

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A032443.

CoefficientList[Series[((1/Sqrt[1-4*x] + 1/(1-4*x))/2)^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 16 2014 *)

G.f. = (g.f. of A032443)^2.
n*a(n) = 32*(2*n-3)*a(n-3) + 48*(1-n)*a(n-2) + 6*(2*n-1)*a(n-1).
Asymptotics: a(n) ~ 2^(2*n)*((n+2)/4 + sqrt(n/Pi)).
Recurrence: (n-2)*n*a(n) = 2*n*(4*n-7)*a(n-1) - 8*(n-1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Apr 16 2014

A318110 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

0, 1, 1, 3, 3, 1, 26, 26, 11, 2, 367, 367, 167, 42, 5, 7142, 7142, 3352, 944, 163, 14, 176766, 176766, 84308, 25006, 4965, 638, 42, 5304356, 5304356, 2554329, 779246, 165474, 24924, 2510, 132, 186954535, 186954535, 90600599, 28120586, 6200455, 1010814, 121086, 9908, 429, 7566084686, 7566084686, 3683084984, 1156456088, 261067596, 44535120, 5829880, 574128, 39203, 1430

Offset: 0

Gheorghe Coserea, Sep 05 2018

			A(x,t) = (1+t)*x + (3+3*t+t^2)*x^2 + (26+26*t+11*t^2+2*t^3)*x^3 + ...
Triangle starts:
n\k [0]       [1]       [2]      [3]      [4]     [5]     [6]    [7]  [8]
[0] 0;
[1] 1,        1;
[2] 3,        3,        1;
[3] 26,       26,       11,      2;
[4] 367,      367,      167,     42,      5;
[5] 7142,     7142,     3352,    944,     163,    14;
[6] 176766,   176766,   84308,   25006,   4965,   638,    42;
[7] 5304356,  5304356,  2554329, 779246,  165474, 24924,  2510,  132;
[8] 186954535,186954535,90600599,28120586,6200455,1010814,121086,9908,429;
[9] ...

Gheorghe Coserea, Rows n=0..100, flattened
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.

Column 0 gives A262301.
Main diagonal gives A000108(n-1) for n>0.
Second diagonal gives A032443(n-1) for n>0.

rows = 10; Clear[A]; A[x_, t_] = (1+t)x;
Do[A[x_, t_] = Series[x t/(1-A[x, t]) + D[A[x, t], t], {x, 0, n}, {t, 0, n}] // Normal, {n, 2 rows}];
CoefficientList[#, t]& /@ CoefficientList[A[x, t], x] /. {} -> {0} // Take[#, rows]& // Flatten (* Jean-François Alcover, Oct 23 2018 *)

seq(N) = {
  my(x='x+O('x^N), t='t, F0=(1+t)*x, F1=0, n=1);
  while(n++,
    F1 = F0^2; F1 = F1 - deriv(F1,'t)/2 + deriv(F0,'t) + x*t;
    if (F1 == F0, break()); F0 = F1);
  concat([[0]], apply(Vecrev, Vec(F0)));
};
concat(seq(10))
\\ test: y=Ser(apply(p->Polrev(p,'t), seq(101)), 'x); y == x*'t/(1-y) + deriv(y,'t)

A(x,t) = Sum_{n>=0} P_n(t)*x^n, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k, satisfies:

A = x*t/(1-A) + deriv(A,t), with A(0,t) = 0, deriv(A,x)(0,t) = 1+t (deriv(A,v) represents the derivative of A with respect to variable v).

A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

Original entry on oeis.org

1, 3, 13, 59, 271, 1250, 5775, 26696, 123423, 570576, 2637306, 12187755, 56312089, 260134905, 1201493926, 5548533913, 25619837773, 118283258215, 546041467522, 2520515546083, 11633752319476, 53693477980816, 247798435809211, 1143547904185879, 5277058908767419

Offset: 0

Seiichi Manyama, Jan 27 2023

nonn

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

Cf. A001700, A032443, A108080, A360144.

Cf. A000108, A000984, A002057.

A360143 := proc(n)
    add(binomial(2*n+2*k,n-k),k=0..n) ;
end proc:
seq(A360143(n),n=0..70) ;# R. J. Mathar, Mar 12 2023

Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 23 2025 *)

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - Stefano Spezia, Jun 17 2025

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

Original entry on oeis.org

1, 1, 68, 22770, 21143488, 41904629550, 151957171590144, 910666718387157732, 8390164064875701321728, 112583179357513548960803670, 2109812207969377622615440752640, 53397692462483465346961668429307836, 1775866125092261344436828225211633500160, 75857512919848315654302238627976991244564300

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

Cf. A032443, A345876, A209289/2, A383853, A383917.

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(3*n), {k, 0, n}], {n, 1, 15}]]

a(n) ~ 2^(2*n + 1/2) * r^(3*n + 1) * n^(3*n) / (sqrt(3 - r^2) * exp(3*n) * (1 - r^2)^n), where r = 0.92488761106894648930384927930334708844525256369797556858640... is the root of the equation (1 + r)/(1 - r) = exp(3/r).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).

Original entry on oeis.org

1, 1, 1028, 14545530, 1127435263168, 309320354959336350, 232325928732003715014144, 403150958104730561230009068564, 1432706082674749593552098155989352448, 9528431104471630510834164178027409070527670, 110580781643902847320855308323644986008860441968640

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

In general, for m>=1, Sum_{k=0..n} binomial(2*n, n-k) * k^(m*n) ~ 2^(2*n + 1/2) * r^(m*n + 1) * n^(m*n) / (sqrt(m + (2-m)*r^2) * exp(m*n) * (1 - r^2)^n), where r is the root of the equation (1 + r)/(1 - r) = exp(m/r).

Cf. A032443 (m=0), A345876 (m=1), A209289/2 (m=2), A383916 (m=3), A383853 (m=4).

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(5*n), {k, 0, n}], {n, 1, 12}]]

a(n) ~ 2^(2*n + 1/2) * r^(5*n + 1) * n^(5*n) / (sqrt(5 - 3*r^2) * exp(5*n) * (1 - r^2)^n), where r = 0.98743428968604456152277643726278132237092161504496484119319... is the root of the equation (1 + r)/(1 - r) = exp(5/r).

A124234 Riordan array (1/(1-x), x(1+x)^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 4, 11, 7, 1, 1, 4, 15, 22, 9, 1, 1, 4, 16, 42, 37, 11, 1, 1, 4, 16, 57, 93, 56, 13, 1, 1, 4, 16, 63, 163, 176, 79, 15, 1, 1, 4, 16, 64, 219, 386, 299, 106, 17, 1, 1, 4, 16, 64, 247, 638, 794, 470, 137, 19, 1

Offset: 0

Paul Barry, Oct 22 2006

Row sums are A077864. Diagonal sums are A004695(n+3). T(2n,n) is A032443.

			Triangle begins
1,
1, 1,
1, 3, 1,
1, 4, 5, 1,
1, 4, 11, 7, 1,
1, 4, 15, 22, 9, 1,
1, 4, 16, 42, 37, 11, 1

Cf. A004695, A032443, A077864.

tabl(nn) = for (n=0, nn, for (k=0, n, print1(sum(j=0, n-k, binomial(2*k, j)), ", ")); print();); \\ Michel Marcus, Nov 05 2016

T(n,k) = Sum_{j=0..n-k} C(2k,j).

A167024 Triangle read by rows: T(n, m) = binomial(n, m)* Sum_{k=0..m} binomial(n, k) for 0 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 21, 8, 1, 20, 66, 60, 16, 1, 30, 160, 260, 155, 32, 1, 42, 330, 840, 855, 378, 64, 1, 56, 609, 2240, 3465, 2520, 889, 128, 1, 72, 1036, 5208, 11410, 12264, 6916, 2040, 256, 1, 90, 1656, 10920, 32256, 48132, 39144, 18072, 4599, 512

Offset: 0

Roger L. Bagula, Oct 27 2009

Row sums are A032443(n).

			1,
1, 2,
1, 6, 4,
1, 12, 21, 8,
1, 20, 66, 60, 16,
1, 30, 160, 260, 155, 32,
1, 42, 330, 840, 855, 378, 64,
1, 56, 609, 2240, 3465, 2520, 889, 128,
1, 72, 1036, 5208, 11410, 12264, 6916, 2040, 256,
1, 90, 1656, 10920, 32256, 48132, 39144, 18072, 4599, 512,
1, 110, 2520, 21120, 81060, 160776, 178080, 116160, 45585, 10230, 1024

Muniru A Asiru, Rows n=0..100 of triangle, flattened

Cf. A008949, A032443.

t:=Flat(List([0..10],n->List([0..n],m->Binomial(n,m)*Sum([0..m],k->Binomial(n,k)))));; Print(t); # Muniru A Asiru, Dec 28 2018

T:=(n, m)-> binomial(n, m)*add(binomial(n, k), k=0..m): seq(seq(T(n, m), m=0..n), n=0..9); # Muniru A Asiru, Dec 28 2018

T[m_, n_] = If[m == 0 && n == 0, 1, Sum[Binomial[m, n]*Binomial[m, k], {k, 0, n}]]
Flatten[Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}]]
T[n_,k_] := Binomial[n, k] (2^n - Binomial[n, k + 1] Hypergeometric2F1[1, 1 -n + k, k + 2, -1]); Table[T[n,k], {n,0,8}, {k,0,n}] // Flatten (* Peter Luschny, Dec 28 2018 *)

T(n, m) = binomial(n,m)*A008949(n,m). [Nov 03 2009]
G.f.: (1/x)*d(arctanh(N(x,y)))/dy, where N(x,y) is g.f. of Narayana numbers (A001263). - Vladimir Kruchinin, Apr 11 2018
T(n, k) = binomial(n, k)*(2^n - binomial(n, 1+k)*hypergeom([1, 1+k-n], [k+2], -1)). - Peter Luschny, Dec 28 2018

Introduced OEIS notational standards in the definition - The Assoc. Editors of the OEIS, Nov 05 2009

A176564 Triangle T(n,m)= binomial(2n,m) + binomial(2n,n-m) -binomial(2*n,n) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, -6, -14, -6, 1, 1, -32, -87, -87, -32, 1, 1, -120, -363, -484, -363, -120, 1, 1, -415, -1339, -2067, -2067, -1339, -415, 1, 1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1, 1, -4844, -16643, -29240, -36992, -36992, -29240

Offset: 0

Roger L. Bagula, Apr 20 2010

Row sums are 1, 2, 4, 4, -24, -236, -1448, -7640, -37424, -175436,... = 2*A032443(n) -A037965(n+1).

			The triangle starts in row n=0 with columns 0<=m<=n as:
1;
1, 1;
1, 2, 1;
1, 1, 1, 1;
1, -6, -14, -6, 1;
1, -32, -87, -87, -32, 1;
1, -120, -363, -484, -363, -120, 1;
1, -415, -1339, -2067, -2067, -1339, -415, 1;
1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1;
1, -4844, -16643, -29240, -36992, -36992, -29240, -16643, -4844, 1;
1, -16776, -58596, -106096, -141151, -153748, -141151, -106096, -58596, -16776, 1;

A176564 := proc(n,m) binomial(2*n,m)+binomial(2*n,n-m) -binomial(2*n,n) ; end proc:

t[n_, m_] = Binomial[2*n, m] + Binomial[2*n, n - m] - (Binomial[2*n, 0] + Binomial[2*n, n]) + 1;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

T(n,m) = T(n,n-m).

A204449 Exponential (or binomial) half-convolution of A000032 (Lucas) with itself.

Original entry on oeis.org

4, 2, 8, 17, 84, 177, 737, 1857, 7732, 19457, 78223, 203777, 809145, 2134017, 8349013, 22347777, 86533892, 234029057, 897748577, 2450784257, 9328491339, 25664946177, 97021416973, 268766806017, 1009936510009, 2814562533377

Offset: 0

Wolfdieter Lang, Jan 16 2012

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576. There the rule for the e.g.f. is also found.

The other half of this exponential half-convolution is found under A204450.

			With A000032 = {2, 1, 3, 4, 7, 11,...}
  a(4) = 1*2*7 + 4*1*4 + 6*3*3 = 84,
  a(5) = 1*2*11 + 5*1*7 + 10*3*4 = 177.

Cf. A000032, 2*A203579 (exponential convolution), A204450.

Table[Sum[Binomial[n, k]*LucasL[k]*LucasL[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 25 2019 *)

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..floor(n/2)), n>=0, with L(n)=A000032(n).
E.g.f.: (l(x)^2 + L2(x^2))/2 with the e.g.f. l(x) of A000032, and the o.g.f. L2(x) of the sequence {(L(n)/n!)^2}.
  l(x)^2 = 2*exp(x)*(cosh(sqrt(5)*x)+1) (see 2*A203579).
  L2(x^2) = BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x), with the golden section phi:=(1+sqrt(5))/2, and for BesselI see Abramowitz-Stegun (reference and link given under A008277), p. 375, eq. 9.6.10.
  BesselI(0,2*sqrt(x)) = hypergeom([],[1],x) is the e.g.f. of {1/n!}.
Bisection: a(2*k) = (2^(2*k)+binomial(2*k,k))*L(2*k)/2 +1 + ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+1, k>=0. For (2^(2*k)+binomial(2*k,k))/2 see A032443(k).

A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2n,kj+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 2, 11, 1, 2, 7, 42, 1, 2, 6, 26, 163, 1, 2, 6, 21, 99, 638, 1, 2, 6, 20, 78, 382, 2510, 1, 2, 6, 20, 71, 297, 1486, 9908, 1, 2, 6, 20, 70, 262, 1145, 5812, 39203, 1, 2, 6, 20, 70, 253, 990, 4447, 22819, 155382, 1, 2, 6, 20, 70, 252, 936, 3796, 17358, 89846, 616666

Offset: 0

Seiichi Manyama, Apr 20 2019

			Square array begins:
      1,    1,    1,    1,    1,    1,    1,    1, ...
      3,    2,    2,    2,    2,    2,    2,    2, ...
     11,    7,    6,    6,    6,    6,    6,    6, ...
     42,   26,   21,   20,   20,   20,   20,   20, ...
    163,   99,   78,   71,   70,   70,   70,   70, ...
    638,  382,  297,  262,  253,  252,  252,  252, ...
   2510, 1486, 1145,  990,  936,  925,  924,  924, ...
   9908, 5812, 4447, 3796, 3523, 3446, 3433, 3432, ...

Columns 1-2 give A032443, A114121.

Cf. A306846, A306915, A307393, A307668.

T[n_, k_] := Sum[Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 13 2021*)

cp's OEIS Frontend

A240879 Self-convolution of Sum(binomial(2*n, i), i=0..n).

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A318110 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

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A383916 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(3*n).

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A383917 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(5*n).

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A124234 Riordan array (1/(1-x), x(1+x)^2).

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A167024 Triangle read by rows: T(n, m) = binomial(n, m)* Sum_{k=0..m} binomial(n, k) for 0 <= m <= n.

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A176564 Triangle T(n,m)= binomial(2*n,m) + binomial(2*n,n-m) -binomial(2*n,n) read by rows.

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A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).

A176564 Triangle T(n,m)= binomial(2n,m) + binomial(2n,n-m) -binomial(2*n,n) read by rows.

A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2n,kj+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.