A084771 -id:A084771 - cp's OEIS frontend

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).

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)

A098659 Expansion of 1/sqrt((1-7x)^2-24x^2).

Original entry on oeis.org

1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695, 11112210781, 126142119187, 1437751935361, 16443380994775, 188609259215725, 2168833084841395, 24994269200292865, 288596644195946695, 3337978523215692925, 38666734085509918675

Offset: 0

Paul Barry, Sep 20 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.
Rob Noble, Asymptotics of a family of binomial sums, J. Number Theory 130 (2010), no. 11, 2561-2585.

Cf. A000984, A001850, A069835, A084771, A084772.

Table[Sum[Binomial[n, k]^2*6^k, {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Sep 15 2012 *)
CoefficientList[Series[1/Sqrt[(1-7*x)^2-24*x^2], {x, 0, 25}], x] (* Stefano Spezia, Dec 04 2018 *)
a[n_] := 5^n*HypergeometricPFQ[{-n,n+1},{1},-1/5]; Table[a[n],{n,0,19}] (* Detlef Meya, May 24 2024 *)

x='x+O('x^66); Vec(1/sqrt(1-14*x+25*x^2)) \\ Joerg Arndt, May 12 2013

G.f.: 1/sqrt(1-14*x+25*x^2).
E.g.f.: exp(7*x)*BesselI(0, 2*sqrt(6)*x).
a(n) = Sum_{k=0..n} C(n, k)^2*6^k.
a(n) = [x^n] (1+7*x+6*x^2)^n.
From Vaclav Kotesovec, Sep 15 2012: (Start)
General recurrence for Sum_{k=0..n} C(n,k)^2*x^k (this is case x=6): (n+2)*a(n+2)-(x+1)*(2*n+3)*a(n+1)+(x-1)^2*(n+1)*a(n)=0.
Asymptotic (Rob Noble, 2010): a(n) ~ (1+sqrt(x))^(2*n+1)/(2*x^(1/4)*sqrt(Pi*n)), this is case x=6. (End)
D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
a(n) = 5^n*hypergeom([-n, n + 1], [1], -1/5). - Detlef Meya, May 24 2024

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200

Offset: 0

Seiichi Manyama, Apr 22 2019

nonn

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..831

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))

{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}

{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

Original entry on oeis.org

1, 3, 22, 245, 3606, 65527, 1411404, 35066313, 985483270, 30869546411, 1065442493556, 40144438269949, 1638733865336764, 72012798200637855, 3388250516614331416, 169894851136173584145, 9041936334960057699654, 508945841697238471315027, 30202327515992972576218980

Offset: 0

Ilya Gutkovskiy, May 31 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A001850, A069835, A084771, A084772, A098659, A187021, A307883, A331656, A335310.

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n,k)^2*(n+1)^k); \\ Michel Marcus, Jun 01 2020

a(n) = central coefficient of (1 + (n + 2)*x + (n + 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 - 2*(n + 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((n + 2)*x) * BesselI(0,2*sqrt(n + 1)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (n+1)^k.
a(n) ~ exp(2*sqrt(n)) * n^(n - 1/4) / (2*sqrt(Pi)) * (1 + 11/(12*sqrt(n))). - Vaclav Kotesovec, Jan 09 2023

A383946 Expansion of 1/sqrt((1-9x)^3 (1-x)).

Original entry on oeis.org

1, 14, 159, 1676, 17005, 168570, 1645035, 15873240, 151863705, 1443272870, 13643264503, 128404376292, 1204055841157, 11255397745298, 104933302809795, 976016662472880, 9059771065058865, 83945271527170110, 776569280469986895, 7173673630527966780, 66182347507155379101, 609866573826736447914

Offset: 0

Seiichi Manyama, Aug 19 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A084771, A331516.

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 9*x)^3 * (1-x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

CoefficientList[Series[1/Sqrt[(1-9*x)^3*(1-x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-9*x)^3*(1-x)))

a(n) = A331516(n) - A331516(n-1).
n*a(n) = (10*n+4)*a(n-1) - 9*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

A387313 Expansion of 1/((1-x) * (1-9*x))^(5/2).

Original entry on oeis.org

1, 25, 415, 5775, 72870, 864150, 9818130, 108109650, 1162302735, 12262882775, 127424209913, 1307536637225, 13276264807260, 133597932407100, 1334029357684980, 13231465264538100, 130461712570627245, 1279632533997010725, 12492837802976030115, 121456026730456739475

Offset: 0

Seiichi Manyama, Aug 25 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A084771, A331516, A387314.

Cf. A387229, A387307.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-9*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/((1-x)*(1-9*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-9*x))^(5/2))

n*a(n) = (10*n+15)*a(n-1) - 9*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-8)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 8^k * 9^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387307(n).
a(n) = (-1)^n * Sum_{k=0..n} 10^k * (9/10)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

A084774 Coefficients of 1/sqrt(1-14x+9x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.

Original entry on oeis.org

1, 7, 69, 763, 8881, 106407, 1298949, 16065483, 200630241, 2524253767, 31947470149, 406281388443, 5187375332881, 66454791792487, 853788052488069, 10996378059281643, 141934540736139201, 1835494145265388167, 23776671158743933509, 308463567293772941883

Offset: 0

Paul D. Hanna, Jun 11 2003

nonn

G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Diagonal of rational functions 1/(1 - x - 2*y - 3*x*y), 1/(1 - x - 2*y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Cf. A006442, A084771.

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*2^k*5^(n-k))); # Muniru A Asiru, Jul 29 2018

[3^n*Evaluate(LegendrePolynomial(n), 7/3) : n in [0..40]]; // G. C. Greubel, May 31 2023

Table[Sum[Binomial[n,k]^2*2^k*5^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
Table[n! SeriesCoefficient[E^(7 x) BesselI[0, 2 Sqrt[10] x], {x,0,n}], {n,0,20}] (* Vincenzo Librandi, May 10 2013 *)
Table[3^n*LegendreP[n, 7/3], {n,0,40}] (* G. C. Greubel, May 31 2023 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, n + 1}, {1}, -2/3]; Flatten[Table[a[n], {n,0,19}]] (* Detlef Meya, May 22 2024 *)

for(n=0,30,t=polcoeff((1+7*x+10*x^2)^n,n,x); print1(t","))

{a(n)=sum(k=0, n, binomial(n, k)^2*2^k*5^(n-k))} \\ Paul D. Hanna, Sep 28 2012

[3^n*gen_legendre_P(n, 0, 7/3) for n in range(41)] # G. C. Greubel, May 31 2023

a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 5^(n-k). - Paul D. Hanna, Sep 28 2012
D-finite with recurrence: n*a(n) = 7*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(200 + 70*sqrt(10))*(7 + 2*sqrt(10))^n/(20*sqrt(Pi*n)) = (sqrt(2) + sqrt(5))^(2*n+1)/(2*10^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 3^n * LegendreP(n, 7/3). - G. C. Greubel, May 31 2023
a(n) = 3^n*hypergeom([-n, n + 1], [1], -2/3). - Detlef Meya, May 22 2024

A341867 Square array read by downward antidiagonals: T(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)binomial(n,j)binomial(i+j,i).

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 8, 12, 12, 8, 16, 28, 33, 28, 16, 32, 64, 86, 86, 64, 32, 64, 144, 216, 245, 216, 144, 64, 128, 320, 528, 664, 664, 528, 320, 128, 256, 704, 1264, 1736, 1921, 1736, 1264, 704, 256, 512, 1536, 2976, 4416, 5322, 5322, 4416, 2976, 1536, 512

Offset: 0

Jianing Song, Nov 07 2021

T(m,n) is the coefficient of x^m*y^n of 1/(1 - 2*x - 2*y + 3*x*y).
In general, define T_{s,t}(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)*binomial(n,j)*binomial(i+j,i)*s^i*t^j, then T_{s,t}(m,n) is the coefficient of x^m*y^n of 1/(1 - (1+s)*x - (1+t)*y + (1+s+t)*x*y).
T(m,n) is the coefficient of x^n of (2 - 3*x)^m/(1 - 2*x)^(m+1). In general, T_{s,t}(m,n) is the coefficient of x^n of ((1+t) - (1+s+t)*x)^m/(1 - (1+s)*x)^(m+1).
T(m,n) is odd if and only if m = n. Proof: T(m,n) == T_{-1,-1}(m,n) (mod 2). The RHS is the coefficient of x^m*y^n of 1/(1 - x*y), which is 1 if m = n and 0 otherwise.
This is the table of cardinalities of bubble posets and shuffle posets, see McConville and Mühle reference. - F. Chapoton, Sep 11 2024

			Rows 0-7:
    1,   2,    4,     8,    16,     32,     64,     128, ...
    2,   5,   12,    28,    64,    144,    320,     704, ...
    4,  12,   33,    86,   216,    528,   1264,    2976, ...
    8,  28,   86,   245,   664,   1736,   4416,   10992, ...
   16,  64,  216,   664,  1921,   5322,  14268,   37272, ...
   32, 144,  528,  1736,  5322,  15525,  43620,  118980, ...
   64, 320, 1264,  4416, 14268,  43620, 127905,  362910, ...
  128, 704, 2976, 10992, 37272, 118980, 362910, 1067925, ...
  ...

Jianing Song, Table of n, a(n) for n = 0..5150 (diagonals 0..100).
T. McConville and H. Mühle, Bubble Lattices I: Structure, arXiv:2202.02874 [math.CO], 2022.
T. McConville and H. Mühle, Bubble Lattices II: Combinatorics, arXiv:2208.13683 [math.CO], 2022.

Cf. A000079 (0th row), A045623(n+1) (1st row), A343561 (2nd row), A084771 (main diagonal).

T[m_, n_] := Sum[Binomial[m, i] * Binomial[n, j] * Binomial[i + j, i], {i, 0, m}, {j, 0, n} ]; Table[T[m, n - m], {n, 0, 9}, {m, 0, n}] // Flatten (* Amiram Eldar, Nov 08 2021 *)
T[m_, n_] := Sum[Binomial[n, j] Hypergeometric2F1[j + 1, -m, 1, -1], {j, 0, n}];
(* Peter Luschny, Nov 08 2021 *)

T(m,n) = sum(i=0, m, sum(j=0, n, binomial(m,i)*binomial(n,j)*binomial(i+j,i)))

T(0,n) = Sum_{k=0..n} binomial(n,k) = 2^n;
T(1,n) = Sum_{k=0..n} binomial(n,k) * (k+2) = (n+4)*2^(n-1);
T(2,n) = Sum_{k=0..n} binomial(n,k) * (k^2+7*k+8)/2 = (n^2+15*n+32)*2^(n-3);
T(3,n) = Sum_{k=0..n} binomial(n,k) * (k^3+15*k^2+56*k+48)/6 = (n^3+33*n^2+254*n+384)*2^(n-4)/3.
E.g.f.: Sum_{m,n>=0} T(m,n)*x^m*y^n/(m!*n!) = exp(2*x+2*y) * BesselI(0,2*sqrt(x*y)). In general, Sum_{m,n>=0} T_{s,t}(m,n)*x^m*y^n/(m!*n!) = exp((1+s)*x+(1+t)*y) * BesselI(0,2*sqrt(s*t*x*y)). Note that BesselI(0,2*sqrt(x)) = Sum_{k>=0} x^k/(k!)^2.
E.g.f. for m-th row: Sum_{n>=0} T(m,n)*x^n/n! = exp(2*x) * Sum_{k=0..m} (binomial(m,k)*2^(m-k)/k!) * x^k. In general, Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * Sum_{k=0..m} (binomial(m,k)*(1+t)^(m-k)/k!) * (s*t*x)^k.
Define P_n(x) = exp(-x) * d^n/dx^n (x^n*exp(x)), then Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * ((1+t)^m/m!) * P_m(s*t*x/(1+t)) if t != -1 and Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * (s*t*x)^m/m! if t = -1.
T(m, n) = Sum_{j=0..n} binomial(n, j)*hypergeom([j + 1, -m], [1], -1). - Peter Luschny, Nov 08 2021

A383600 Expansion of 1/( (1-x)^3 * (1-9*x) )^(1/4).

Original entry on oeis.org

1, 3, 15, 97, 699, 5313, 41689, 334215, 2721411, 22423737, 186497325, 1562826195, 13178010405, 111700773135, 951026829255, 8128169277897, 69701329848051, 599462375836185, 5169038197383789, 44674793959777443, 386916485124220929, 3357265884164614707

Offset: 0

Seiichi Manyama, May 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A084771, A383602.

Cf. A004981, A231482.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x)^3 * (1-9*x) )^(1/4))); // Vincenzo Librandi, May 05 2025

Table[Sum[(-8)^(k)* Binomial[-1/4,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)

a(n) = sum(k=0, n, (-8)^k*binomial(-1/4, k)*binomial(n, k));

a(n) = Sum_{k=0..n} (-8)^k * binomial(-1/4,k) * binomial(n,k).
n*a(n) = (10*n-7)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.
a(n) ~ 3^(2*n + 3/2) / (Gamma(1/4) * 2^(9/4) * n^(3/4)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([1/4, -n], [1], -8). - Stefano Spezia, May 05 2025

A383602 Expansion of 1/( (1-x) * (1-9*x)^3 )^(1/4).

Original entry on oeis.org

1, 7, 55, 453, 3819, 32637, 281409, 2441715, 21285411, 186225253, 1633973125, 14370441055, 126631522005, 1117707358515, 9879287145855, 87428272217853, 774533435844531, 6868083093333285, 60952616213098789, 541342619512077967, 4811079933571973329

Offset: 0

Seiichi Manyama, May 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A084771, A383600.

Cf. A004982.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x) * (1-9*x)^3 )^(1/4))); // Vincenzo Librandi, May 05 2025

Table[Sum[(-8)^k* Binomial[-3/4,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)

a(n) = sum(k=0, n, (-8)^k*binomial(-3/4, k)*binomial(n, k));

a(n) = Sum_{k=0..n} (-8)^k * binomial(-3/4,k) * binomial(n,k).
n*a(n) = (10*n-3)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.
a(n) ~ Gamma(1/4) * 3^(2*n + 1/2) / (Pi * 2^(5/4) * n^(1/4)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([3/4, -n], [1], -8). - Stefano Spezia, May 05 2025

cp's OEIS Frontend

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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A098659 Expansion of 1/sqrt((1-7*x)^2-24*x^2).

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A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

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A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

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A383946 Expansion of 1/sqrt((1-9*x)^3 * (1-x)).

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A387313 Expansion of 1/((1-x) * (1-9*x))^(5/2).

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A084774 Coefficients of 1/sqrt(1-14*x+9*x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.

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A098659 Expansion of 1/sqrt((1-7x)^2-24x^2).

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).

A383946 Expansion of 1/sqrt((1-9x)^3 (1-x)).

A084774 Coefficients of 1/sqrt(1-14x+9x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.

A341867 Square array read by downward antidiagonals: T(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)binomial(n,j)binomial(i+j,i).