A026375 -id:A026375 - cp's OEIS frontend

A293490 a(n) = Sum_{k=0..n} binomial(2k, k)binomial(2*n-k, n).

1, 4, 18, 84, 400, 1932, 9436, 46512, 231066, 1155660, 5813808, 29396952, 149305884, 761282032, 3894953640, 19987999696, 102847396416, 530446714812, 2741576339716, 14196136939600, 73631851898220, 382483602131400, 1989514312826400, 10361255764532400, 54020655931542300, 281933439875693424

Offset: 0

Ilya Gutkovskiy, Oct 10 2017

nonn

Main diagonal of iterated partial sums array of central binomial coefficients (starting with the first partial sums).

Cf. A000984, A002426, A006134, A026375, A270447.

A293490 := Concatenation([1], List([1..3*10^2],n -> Sum([0..n],k -> Binomial(2*k, k)*(Binomial(2*n - k, n))))); # Muniru A Asiru, Oct 15 2017

Table[Sum[Binomial[2 k, k] Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/(Sqrt[1 - 4 x] (1 - x)^(n + 1)), {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 - 2 x/(1 + ContinuedFractionK[-x, 1, {k, 1, n}])), {x, 0, n}], {n, 0, 25}]
CoefficientList[Series[1/(Sqrt[2 Sqrt[1-4 x]-1] Sqrt[1-4 x]),{x,0,25}],x] (* Alexander M. Haupt, Jun 24 2018 *)

a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k, n)); \\ Michel Marcus, Oct 15 2017

a(n) = [x^n] 1/(sqrt(1 - 4*x)*(1 - x)^(n+1)).
a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))), a continued fraction.
a(n) = 4^n*Gamma(n+1/2)*2F1(-n,n+1; 1/2-n; 1/4)/(sqrt(Pi)*Gamma(n+1)).
From Vaclav Kotesovec, Oct 16 2017: (Start)
D-finite with recurrence: 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n + 3/2) / (3^(n + 1/2) * sqrt(Pi*n)).
(End)
G.f.: 1/(sqrt(2*sqrt(1-4*x)-1)*sqrt(1-4*x)). - Alexander M. Haupt, Jun 24 2018

A383254 Expansion of 1/sqrt( (1-x) * (1-5*x)^3 ).

Original entry on oeis.org

1, 8, 51, 300, 1695, 9348, 50729, 272128, 1447155, 7643880, 40156281, 210019428, 1094338401, 5684293020, 29446107975, 152181330480, 784880109315, 4040712839880, 20768844586025, 106595697483700, 546389531720445, 2797395801163260, 14306735857573995

Offset: 0

Seiichi Manyama, May 05 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A383499, A383503.

Cf. A026375.

[&+[(2*k+1)*Binomial(2*k, k)*Binomial(n+1, k+1): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2025

Table[Sum[(2*k+1)* Binomial[2*k, k]*Binomial[n+1,k+1],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 18 2025 *)

a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+1, k+1));

a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+1,k+1).
D-finite with recurrence n*a(n) +2*(-3*n-1)*a(n-1) +5*n*a(n-2)=0. - R. J. Mathar, May 05 2025
a(n) ~ 5^(n + 1/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 05 2025
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k) * binomial(n+1,n-k). (End)

A110166 Row sums of Riordan array A110165.

Original entry on oeis.org

1, 4, 18, 85, 410, 1999, 9807, 48304, 238570, 1180615, 5851253, 29033074, 144190943, 716652070, 3564079250, 17734184365, 88280673770, 439625873215, 2189988826125, 10912480440850, 54389237971285, 271142650382080

Offset: 0

Paul Barry, Jul 14 2005

Number of 5-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012
From Peter Bala, Jan 09 2022: (Start)
Conjectures: for k >= 2, the number of k-ary words of length n such that the number of 1's <= the number of 0's is equal to the coefficient of x^n in the expansion of ( k*x + 1/(1 + x) )^n, and satisfies the recurrence u(0) = 1, u(1) = k-1 and n*u(n) = (k-2)*(2*n-1)*u(n-1) - k*(k-4)*(n-1)* u(n-2) + k^(n-1) for n >= 2.
For cases see A027306 (k = 2), A027914 (k = 3) and A032443 (k = 4). (End)

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

Row sums A110165. Cf. A026375, A032443, A246437, A242586.

seq( (1/2)*(5^n + add(binomial(n,k)*binomial(2*k,k), k = 0..n)), n = 0..30); # Peter Bala, Jan 08 2022

Table[Sum[Sum[Binomial[n,j]Binomial[2j,j+k],{j,0,n}],{k,0,n}],{n,0,25}] (* Harvey P. Dale, Dec 16 2011 *)

G.f.: (1/sqrt(1-6*x+5*x^2))/(1-(1-3*x-sqrt(1-6*x+5*x^2))/(2*x)).
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} C(n, j)*C(2*j, j+k).
Recurrence: n*a(n) = (11*n-8)*a(n-1) - 5*(7*n-10)*a(n-2) + 25*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 5^n/2*(1+sqrt(5)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Oct 18 2012
From Peter Bala, Jan 08 2022: (Start)
a(n) = (1/2)*(5^n + A026375(n)) = (1/2)*(5^n + Sum_{k = 0..n} binomial(n,k) *binomial(2*k,k)).
a(n) = (1/2)*(5^n)*(1 + Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(-1/5)^k).
a(n) = [x^n] ( 5*x + 1/(1 + x) )^n.
a(0) = 1, a(1) = 4 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) + 5^(n-1) for n >= 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
Binomial transform of A032443. (End)

A171128 A117852*A130595 as lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 19, 28, 18, 4, 1, 51, 95, 70, 30, 5, 1, 141, 306, 285, 140, 45, 6, 1, 393, 987, 1071, 665, 245, 63, 7, 1, 1107, 3144, 3948, 2856, 1330, 392, 84, 8, 1, 3139, 9963, 14148, 11844, 6426, 2394, 588, 108, 9, 1

Offset: 0

Philippe Deléham, Dec 04 2009

Mirror image of triangle in A135091.

Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - Peter Bala, Feb 12 2017

			Triangle begins:
   1
   1  1
   3  2  1
   7  9  3 1
  19 28 18 4 1
  ...
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator begins
      0
      1    0
      2    2     0
      0    6     3     0
     -6    0    12     4     0
      0  -30     0    20     5   0
     80    0   -90     0    30   6   0
      0  560     0  -210     0  42   7  0
  -2310    0  2240     0  -420   0  56  8  0
  ....
and equals the generalized exponential Riordan array [x + log(Bessel_I(0,2*x)), x], and so has integer entries. (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

A000984 (row sums), A135091 (row reversed). Cf. A002426, A117852, A130595, A109187.

A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[ Binomial[n, k]*A002426[n - k], {n, 0, 99}, {k, 0, n} ]//Flatten (* G_. C. Greubel_, Mar 07 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n), A104454(n) for x = -1,0,1,2,3,4,5,6 respectively.
T(n,k) = binomial(n,k)*A002426(n-k). - Philippe Deléham, Dec 12 2009
From Peter Bala, Feb 12 2017: (Start)
T(n,k) = Sum_{j = 0..floor((n-k)/2)} n!/((n-k-2*j)!*j!^2*k!).
T(n,k) = n/k*T(n-1,k-1) with T(n,0) = A002426(n).
(n - k)^2*T(n,k) = n*(2*n - 2*k - 1)*T(n-1,k) + 3*n*(n - 1)*T(n-2,k).
O.g.f. = 1/sqrt((1 - (1 + t)*z)^2 - 4*z^2) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2 + (7 + 9*t + 3*t^2 + t^3 )*z^3 + ....
E.g.f. Bessel_I(0,2*x) * exp((1 + t)*x) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2/2! + (7 + 9*t + 3*t^2 + t^3 )*z^3/3! + ....
n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)*(1 + t)^(n-2*k) = coefficient of x^n in the expansion of (1 + (1 + t)*x + x^2)^n.
The polynomials R(n, t - 1) are the row polynomials of A109187.
d/dt(R(n,t)) = n*R(n-1,t).
Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-1 .. t+3} x^n/sqrt((t + 3 - x)*(x - t + 1)) dx.
The zeros of the row polynomials appear to lie on the vertical line Re(z) = -1 in the complex plane, and the zeros of R(n,t) and R(n+1,t) appear to interlace along this line.
(End)

A248168 Expansion of g.f. 1/sqrt((1-3x)(1-11*x)).

Original entry on oeis.org

1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, 518839783, 5414767897, 56795795679, 598213529809, 6322787125207, 67026654455433, 712352213507151, 7587639773475777, 80977812878889927, 865716569022673401, 9269461606674304959, 99387936492243451569, 1066975862517563301303

Offset: 0

Paul D. Hanna, Oct 03 2014

nonn

			G.f.: A(x) = 1 + 7*x + 57*x^2 + 511*x^3 + 4849*x^4 + 47607*x^5 +...
where A(x)^2 = 1/((1-3*x)*(1-11*x)):
A(x)^2 = 1 + 14*x + 163*x^2 + 1820*x^3 + 20101*x^4 + 221354*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..961
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Cf. A248167, A084771.

[n le 2 select 7^(n-1) else (7*(2*n-3)*Self(n-1) - 33*(n-2)*Self(n-2))/(n-1) : n in [1..40]]; // G. C. Greubel, May 31 2025

CoefficientList[Series[1/Sqrt[(1-3*x)*(1-11*x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

{a(n)=polcoeff( 1 / sqrt((1-3*x)*(1-11*x) +x*O(x^n)), n) }
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff( (1 + 7*x + 4*x^2 +x*O(x^n))^n, n) }
for(n=0, 25, print1(a(n), ", "))

{a(n)=sum(k=0,n, 3^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0, 25, print1(a(n), ", "))

@CachedFunction
def A248168(n):
     if (n<2): return 7^n
     else: return (7*(2*n-1)*A248168(n-1) - 33*(n-1)*A248168(n-2))//n
print([A248168(n) for n in range(41)]) # G. C. Greubel, May 31 2025

a(n) equals the central coefficient in (1 + 7*x + 4*x^2)^n, n>=0.
a(n) = Sum_{k=0..n} 3^(n-k) * 2^k * C(n,k) * C(2*k,k).
a(n) = Sum_{k=0..n} 11^(n-k) * (-2)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Apr 20 2019
a(n)^2 = A248167(n), which gives the coefficients in 1 / AGM(1-3*11*x, sqrt((1-3^2*x)*(1-11^2*x))).
Equals the binomial transform of 2^n*A026375(n).
Equals the second binomial transform of A084771.
Equals the third binomial transform of A059304(n) = 2^n*(2*n)!/(n!)^2.
a(n) ~ 11^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 03 2014
D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +33*(n-1)*a(n-2)=0. [Belbachir]
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A340970 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(2*j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 1, 7, 33, 45, 1, 1, 9, 67, 245, 195, 1, 1, 11, 113, 721, 1921, 873, 1, 1, 13, 171, 1593, 8179, 15525, 3989, 1, 1, 15, 241, 2981, 23649, 95557, 127905, 18483, 1, 1, 17, 323, 5005, 54691, 361449, 1137709, 1067925, 86515, 1

Offset: 0

Seiichi Manyama, Feb 01 2021

			Square array begins:
  1,   1,     1,     1,      1,       1, ...
  1,   3,     5,     7,      9,      11, ...
  1,  11,    33,    67,    113,     171, ...
  1,  45,   245,   721,   1593,    2981, ...
  1, 195,  1921,  8179,  23649,   54691, ...
  1, 873, 15525, 95557, 361449, 1032801, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000012, A026375, A084771, A340973.
Rows n=0..2 give A000012, A005408, A080859.
Main diagonal gives A340971.
Cf. A340968.

T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * Binomial[2*j, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)

T(n, k) = sum(j=0, n, k^j*binomial(n, j)*binomial(2*j, j));

T(n, k) = polcoef((1+(2*k+1)*x+(k*x)^2)^n, n);

G.f. of column k: 1/sqrt((1 - x) * (1 - (4*k+1)*x)).
T(n,k) = [x^n] (1+(2*k+1)*x+(k*x)^2)^n.
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0,2*k*x). - Ilya Gutkovskiy, Feb 01 2021
From Seiichi Manyama, Aug 19 2025: (Start)
T(n,k) = (1/4)^n * Sum_{j=0..n} (4*k+1)^j * binomial(2*j,j) * binomial(2*(n-j),n-j).
T(n,k) = Sum_{j=0..n} (-k)^j * (4*k+1)^(n-j) * binomial(n,j) * binomial(2*j,j). (End)

A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

Original entry on oeis.org

1, 5, 37, 317, 2885, 27105, 259765, 2523813, 24768069, 244941833, 2437083697, 24367722725, 244639635749, 2464477467769, 24899468129405, 252202062544617, 2560119328830725, 26038134699958233, 265278657849511561, 2706809063101138409, 27657194997231516145, 282941098708193905485

Offset: 0

Vaclav Kotesovec, Jan 09 2023

nonn

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

Robert Israel, Table of n, a(n) for n = 0..977

Cf. A026375, A188686.

Cf. A156887, A346646, A346664.

A359643 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[1/3,2/3,1],-256/27) ;
    simplify(%) ;
end proc:
seq(A359643(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Table[Sum[Binomial[n, k]*Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(4*k,k)); \\ Michel Marcus, Jan 09 2023

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).
Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 09 2023
a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - R. J. Mathar, Jan 10 2023
a(n) = [x^n] (1 + 5*x + 6*x^2 + 4*x^3 + x^4)^n. - Ilya Gutkovskiy, Apr 17 2025

A383573 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(2(n-2k),n-2*k).

Original entry on oeis.org

1, 2, 7, 24, 89, 338, 1311, 5152, 20449, 81778, 328999, 1330008, 5398265, 21984610, 89791103, 367643776, 1508560257, 6201927074, 25540266503, 105336838616, 435035342553, 1798875915826, 7446653956895, 30857577536800, 127987031688161, 531301328367762, 2207281722474919

Offset: 0

Seiichi Manyama, Apr 30 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A026375, A383581, A383582.

Cf. A360290.

[&+[Binomial(n-k, k) * Binomial(2*(n-2*k), n-2*k): k in [0..Floor(n div 2)]]: n in [0..35]]; // Vincenzo Librandi, May 03 2025

Table[Sum[Binomial[n-k,k]* Binomial[2*(n-2*k),n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 03 2025 *)

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

G.f.: 1/sqrt((1 - x^2) * (1 - x^2 - 4*x)).

a(n) ~ phi^(3*n + 3) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 01 2025

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-3k),n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 74, 270, 1005, 3788, 14418, 55289, 213270, 826614, 3216629, 12558928, 49175136, 193023965, 759299438, 2992534344, 11813985377, 46709675040, 184928644350, 733047010709, 2908981549006, 11555513379450, 45945148281437, 182835149061920, 728149606630164

Offset: 0

Seiichi Manyama, Apr 30 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A026375, A383573, A383582.
Cf. A376791, A383571.
Cf. A360291.

[&+[Binomial(n-2*k,k) * Binomial(2*(n-3*k),n-3*k): k in [0..n div 3]]: n in [0..25]]; // Vincenzo Librandi, May 02 2025

Table[Sum[Binomial[n-2*k,k]* Binomial[2*(n-3*k),n-3*k],{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)

a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(2*(n-3*k), n-3*k));

G.f.: 1/sqrt((1 - x^3) * (1 - x^3 - 4*x)).

A383582 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3k,k) binomial(2(n-4k),n-4*k).

Original entry on oeis.org

1, 2, 6, 20, 71, 256, 942, 3512, 13221, 50138, 191260, 733088, 2821037, 10892100, 42174848, 163706656, 636816019, 2481902842, 9689155902, 37882580356, 148313102097, 581365577564, 2281393560802, 8961689897248, 35235582858441, 138657185501870, 546064549476476

Offset: 0

Seiichi Manyama, Apr 30 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A026375, A383573, A383581.
Cf. A376792, A383572.
Cf. A360292.

[&+[Binomial(n-3*k,k) * Binomial(2*(n-4*k),n-4*k): k in [0..n div 4]]: n in [0..45]]; // Vincenzo Librandi, May 02 2025

Table[Sum[Binomial[n-3*k,k]* Binomial[2*(n-4*k),n-4*k],{k,0,Floor[n/4]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)

a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-4*k), n-4*k));

G.f.: 1/sqrt((1 - x^4) * (1 - x^4 - 4*x)).

a(n) ~ (2 + sqrt(2) + sqrt(10 + 8*sqrt(2)))^n / (sqrt((sqrt(5 + 32*sqrt(2)) - 7)*Pi*n) * 2^(n + 7/4)). - Vaclav Kotesovec, May 01 2025

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A293490 a(n) = Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-k, n).

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GAP

Mathematica

PARI

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A383254 Expansion of 1/sqrt( (1-x) * (1-5*x)^3 ).

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Magma

Mathematica

PARI

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A110166 Row sums of Riordan array A110165.

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Maple

Mathematica

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A171128 A117852*A130595 as lower triangular matrices.

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A248168 Expansion of g.f. 1/sqrt((1-3*x)*(1-11*x)).

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Magma

Mathematica

PARI

PARI

PARI

SageMath

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A340970 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(2*j,j).

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PARI

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A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

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A293490 a(n) = Sum_{k=0..n} binomial(2k, k)binomial(2*n-k, n).

A248168 Expansion of g.f. 1/sqrt((1-3x)(1-11*x)).

A383573 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(2(n-2k),n-2*k).

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-3k),n-3*k).

A383582 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3k,k) binomial(2(n-4k),n-4*k).