A069835 -id:A069835 - cp's OEIS frontend

A098269 a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.

1, 8, 94, 1232, 16966, 240368, 3468844, 50712992, 748553926, 11131168688, 166498969924, 2502416381792, 37759888297756, 571681667171168, 8679980422677784, 132116085646644032, 2015249400937940806

Offset: 0

Paul Barry, Sep 01 2004

Central coefficients of (1+8x+15x^2)^n. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2).
16th binomial transform of 2^n*LegendreP(n,-4) = (-1)^n*A098269(n). - Paul Barry, Sep 03 2004
Diagonal of rational functions 1/(1 + x + 3*y + x*z - 2*x*y*z), 1/(1 - x + y + 3*x*z - 2*x*y*z), 1/(1 - x - x*y - 3*y*z - 2*x*y*z). - Gheorghe Coserea, Jul 03 2018

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.

Cf. A069835, A084773.

Table[SeriesCoefficient[1/Sqrt[1-16*x+4*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n}, {1}, 5/3]; Flatten[Table[a[n], {n,0,16}]] (* Detlef Meya, May 21 2024 *)

a(n)=pollegendre(n,4)<Charles R Greathouse IV, Oct 24 2011

{a(n)=sum(k=0, n, binomial(n, k)^2*3^k*5^(n-k))} \\ Paul D. Hanna, Sep 29 2012

G.f.: 1/sqrt(1-16x+4x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, k)*binomial(2(n-k), n)*4^(n-2k).
E.g.f.: exp(8*x)*BesselI(0, 2*sqrt(15)*x), cf. A084770. - Vladeta Jovovic, Sep 01 2004
a(n) = Sum_{k=0..n} binomial(n,k)^2 * 3^k * 5^(n-k). - Paul D. Hanna, Sep 29 2012
D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(450+120*sqrt(15))*(8+2*sqrt(15))^n/(30*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 3^n*hypergeom([-n, -n], [1], 5/3) = 5^n*hypergeom([-n, -n], [1], 3/5). - Detlef Meya, May 21 2024

A110124 A scaled Legendre triangle.

Original entry on oeis.org

1, 0, 1, -2, 2, 1, 0, 4, 4, 1, 6, 8, 22, 6, 1, 0, 16, 136, 52, 8, 1, -20, 32, 886, 504, 94, 10, 1, 0, 64, 5944, 5136, 1232, 148, 12, 1, 70, 128, 40636, 53856, 16966, 2440, 214, 14, 1, 0, 256, 281488, 575296, 240368, 42256, 4248, 292, 16, 1, -252, 512, 1968934, 6225792, 3468844, 752800, 88566, 6776, 382, 18, 1

Offset: 0

Paul Barry, Jul 13 2005

Row sums are A110125. Diagonal sums are A110126. Columns include A000079, A069835, A084773, and A098269.

			Rows begin
1;
0,1;
-2,2,1;
0,4,4,1;
6,8,22,6,1;
0,16,136,62,8,1;
-20,32,886,504,94,10,1;

T(n, k)=pollegendre(n-k, k)<<(n-k) \\ Charles R Greathouse IV, Mar 18 2017

Number triangle T(n, k)=2^(n-k)*LegendreP(n-k, k); T(n, k)=sum{j=0..floor((n-k)/2), (-1)^j*C(n-k, j)C(2n-2k-2j, n-k)k^(n-k-2j)}.

A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 7, 15, 9, 1, 8, 22, 24, 9, 1, 11, 46, 90, 81, 27, 1, 12, 57, 136, 171, 108, 27, 1, 15, 93, 307, 579, 621, 351, 81, 1, 16, 108, 400, 886, 1200, 972, 432, 81, 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243, 1, 20, 175, 880, 2810, 5944, 8430, 7920

Offset: 0

Philippe Deléham, Dec 14 2008

			The triangle T(n,k) begins:
n\k  0   1    2     3     4      5      6      7      8      9     10    11   12
0:   1
1:   1   3
2:   1   4    3
3:   1   7   15     9
4:   1   8   22    24     9
5:   1  11   46    90    81     27
6:   1  12   57   136   171    108     27
7:   1  15   93   307   579    621    351     81
8:   1  16  108   400   886   1200    972    432     81
9:   1  19  156   724  2086   3858   4572   3348   1377    243
10:  1  20  175   880  2810   5944   8430   7920   4725   1620    243
11:  1  23  235  1405  5450  14374  26262  33210  28485  15795   5103   729
12:  1  24  258  1640  6855  19824  40636  59472  61695  44280  20898  5832  729
... reformatted and extended. - _Franck Maminirina Ramaharo_, Feb 28 2018

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf. A152815, A007318, A064861.

a152842 n k = a152842_tabl !! n !! k
a152842_row n = a152842_tabl !! n
a152842_tabl = map fst $ iterate f ([1], 3) where
   f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 4 - z)
-- Reinhard Zumkeller, May 01 2014

T(n,k) = T(n-1,k) + (2-(-1)^n)*T(n-1,k-1).
Sum_{k=0..n} T(n,k) = A094015(n).
T(n,n) = A108411(n+1).
T(2n,n) = A069835(n).
G.f.: (1+x+x*y)/(1-x^2-4*x^2*y-3*x^2*y^2). - Philippe Deléham , Nov 09 2013
T(n,k) = T(n-2,k) + 4*T(n-2,k-1) + 3*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

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

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

A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

Original entry on oeis.org

1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786

Offset: 0

Seiichi Manyama, Aug 27 2025

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

Cf. A069835, A331792, A387339.

Cf. A002696, A387338.

[&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.

A054122 T(2n,n), array T as in A054120.

Original entry on oeis.org

1, 3, 18, 114, 750, 5058, 34692, 240852, 1687446, 11906610, 84490428, 602335836, 4310670156, 30950436468, 222844189320, 1608369633384, 11632913018598, 84294762019218, 611831430746124, 4447397950230540

Offset: 0

Clark Kimberling

nonn

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

Cf. A069835.

Table[SeriesCoefficient[(1-x)/Sqrt[1-8*x+4*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)

a(n)=(2*pollegendre(n,2) - pollegendre(n-1,2))<<(n-1) \\ Charles R Greathouse IV, Mar 18 2017

a(n) = A069835(n) - A069835(n-1).
G.f.: (1-x)/sqrt(1-8*x+4*x^2). - Vladeta Jovovic, May 13 2003
D-finite with recurrence: (n-1)*n*a(n) = 4*(n-1)*(2*n-1)*a(n-1) - 4*(n-2)*n*a(n-2). - Vaclav Kotesovec, Oct 09 2012
a(n) ~ sqrt(6+4*sqrt(3))*(4+2*sqrt(3))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 09 2012
D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +12*(n-2)*a(n-2) +4*(-n+3)*a(n-3)=0. - R. J. Mathar, Jun 13 2013

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

Original entry on oeis.org

1, 3, 15, 99, 711, 5373, 42099, 338355, 2771127, 23028813, 193610385, 1643215005, 14056350075, 121040308665, 1048212778635, 9122168556819, 79727173530327, 699443806767525, 6156776010386481, 54356715121718349, 481194980656865721, 4270165015550478003

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 3*x*y*z), 1/(1 - x*y + y*z + x*z - 3*x*y*z). - Gheorghe Coserea, Jul 04 2018
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + 3*x*y*z)). - Seiichi Manyama, Jul 05 2025

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
where
A(x) = 1/(1-3*x) + 6*x^2/(1-3*x)^4 + 90*x^4/(1-3*x)^7 + 1680*x^6/(1-3*x)^10 + 34650*x^8/(1-3*x)^13 + 756756*x^10/(1-3*x)^16 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Cf. A000172, A002893, A069835, A208425, A294035.

Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-3*x+x*O(x^n))^(3*m+1)),n)}
for(n=0,31,print1(a(n),", "))

a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ Gheorghe Coserea, Jul 04 2018

From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 9*x - 1)*y'' + (243*x^4 + 216*x^3 + 27*x^2 + 36*x - 2)*y' + 3*(27*x^3 + 33*x^2 - 2*x + 2)*y.
(End)
From Vaclav Kotesovec, Oct 07 2020: (Start)
Recurrence: n^2*(3*n - 5)*a(n) = 3*(9*n^3 - 24*n^2 + 17*n - 4)*a(n-1) + 3*(3*n - 4)*a(n-2) + 27*(n-2)^2*(3*n - 2)*a(n-3).
a(n) ~ sqrt(2 + sqrt(5)*phi^(-1/3) + sqrt(5)*phi^(1/3)) * 3^n * (1 + phi^(-2/3) + phi^(2/3))^n / (2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
(End)

A382405 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^(n-k).

Original entry on oeis.org

1, 4, 34, 352, 4006, 48184, 600916, 7687936, 100240198, 1326277144, 17753591164, 239915864896, 3267780399196, 44805617380528, 617844108170344, 8561667414341632, 119151750609504838, 1664497333624420888, 23330380347342383404, 327990673915214512192, 4623496960858710060916

Offset: 0

Ilya Gutkovskiy, Apr 08 2025

Diagonal of the rational function 1 / (1 - x - x*y - y*z - 2*x*z - 2*x*y*z).

Cf. A001850, A005258, A069835, A274671, A382642, A382848.

seq(simplify(2^n*hypergeom([-n, -n, n+1], [1, 1], 1/2)), n = 0..20); # Peter Bala, May 23 2025

Table[Sum[Binomial[n, k]^2 Binomial[n + k, k] 2^(n - k), {k, 0, n}], {n, 0, 20}]
Table[2^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, 1/2], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - x - x y - y z - 2 x z - 2 x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 20}]

a(n) = 2^n * hypergeom([-n, -n, n+1], [1, 1], 1/2).
From Peter Bala, May 23 2025: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(n, k)*binomial(2*k, n)^2.
(11*n - 16)*n^2*a(n) = 2*(77*n^3 - 189*n^2 + 132*n - 30)*a(n-1) + 4*(33*n^3 - 114*n^2 + 124*n - 40)*a(n-2) + 4*(11*n - 5)*(n - 2)^2*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 34. (End)
a(n) ~ sqrt((55 + (22*(7513 - 183*sqrt(33)))^(1/3) + (22*(7513 + 183*sqrt(33)))^(1/3)) / 33) * ((14 + (1/3)*(95958 - 1782*sqrt(33))^(1/3) + (2*(1777 + 33*sqrt(33)))^(1/3)) / 3)^n / (2*Pi*n). - Vaclav Kotesovec, Jun 07 2025

A387343 Expansion of 1/(1 - 8x + 4x^2)^(5/2).

Original entry on oeis.org

1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396

Offset: 0

Seiichi Manyama, Aug 27 2025

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

Cf. A069835, A331515, A387344.
Cf. A002802, A387313, A387341.
Cf. A387339.

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))

n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387339(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).

cp's OEIS Frontend

A098269 a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.

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A110124 A scaled Legendre triangle.

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A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Haskell

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

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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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A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

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Magma

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A054122 T(2n,n), array T as in A054120.

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A208426 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-3*x)^(3*n+1).

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

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

A387343 Expansion of 1/(1 - 8x + 4x^2)^(5/2).