A026375 -id:A026375 - cp's OEIS frontend

A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).

1, 15, 145, 1155, 8260, 55188, 351960, 2170080, 13042095, 76827465, 445335891, 2547479025, 14412134100, 80773641900, 449065521300, 2479190589180, 13603361708775, 74238475926825, 403197150223175, 2180369322394725, 11744998515662720, 63044308615576200, 337323759106291100

Offset: 0

Seiichi Manyama, Aug 23 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026375, A385563, A387238.

Cf. A026377, A374508.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[1/((1-x)*(1-5*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

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

n*a(n) = (6*n+9)*a(n-1) - 5*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k) = (binomial(n+4,2)/6) * A026377(n+2).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

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

A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

Original entry on oeis.org

1, 21, 266, 2646, 22806, 178794, 1310694, 9140274, 61330269, 399107709, 2533330800, 15751925280, 96257031780, 579556206180, 3445117599480, 20252115155160, 117890464642335, 680320688005035, 3895668955041710, 22152779612619810, 125183331416173030

Offset: 0

Seiichi Manyama, Aug 23 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026375, A385563, A387237.

Cf. A126190, A374509, A387239.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[1/((1-x)*(1-5*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(7/2))

n*a(n) = (6*n+15)*a(n-1) - 5*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387239(n).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).

A026380 a(n) = T(n,[ n/2 ]), where T is the array in A026374.

Original entry on oeis.org

1, 3, 4, 11, 17, 45, 75, 195, 339, 873, 1558, 3989, 7247, 18483, 34016, 86515, 160795, 408105, 764388, 1936881, 3650571, 9238023, 17501619, 44241261, 84179877, 212601015, 406020930, 1024642875, 1963073865, 4950790605

Offset: 0

Clark Kimberling

nonn

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012

Cf. A026375, A026378.

a(2n)=A026378(n+1), a(2n-1)=A026375(n). - Emeric Deutsch, Feb 18 2004
a(2n) = A026378(2n+1), a(2n+1) = A026375(n+1).
Davenport et al. give a g.f.

A117852 Mirror image of A098473 formatted as a triangular array.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 18, 6, 1, 70, 80, 36, 8, 1, 252, 350, 200, 60, 10, 1, 924, 1512, 1050, 400, 90, 12, 1, 3432, 6468, 5292, 2450, 700, 126, 14, 1, 12870, 27456, 25872, 14112, 4900, 1120, 168, 16, 1, 48620, 115830, 123552, 77616, 31752, 8820, 1680, 216, 18, 1

Offset: 0

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

			Triangle begins:
    1;
    2,   1;
    6,   4,   1;
   20,  18,   6,   1;
   70,  80,  36,   8,   1;
  252, 350, 200,  60,  10,   1;
  ...

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

Cf. A098473.

c:=n->binomial(2*n, n): T:=proc(n, k) if k<=n then binomial(n, k)*c(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; #

Table[ Binomial[n, k]*Binomial[2*n - 2*k, n - k], {n,0,10}, {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) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
T(n,k) = binomial(n,k)*A000984(n-k). - Philippe Deléham, Dec 12 2009
O.g.f.:  1/sqrt( (1 - x*t)*(1 - (x + 4)*t) ) = 1 + (2 + x)*t + (6 + 4*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 12 2007

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.

Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Cf. A061554, A158793, A158815, A171224.

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.

A242586 Expansion of 1/(2sqrt(1-x))(1/sqrt(1-x)+1/(sqrt(1-5*x))).

Original entry on oeis.org

1, 2, 6, 23, 98, 437, 1995, 9242, 43258, 204053, 968441, 4619012, 22120631, 106300508, 512321438, 2475395303, 11986728458, 58156146653, 282640193313, 1375737276788, 6705522150973, 32724071280518, 159878425878848

Offset: 0

Vladimir Kruchinin, May 18 2014

Binomial transform of A088218.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007317, A088218, A105477, A110166, A246437.

a := n -> hypergeom([1/2,-n],[1], -4)/2 + 1/2;
seq(round(evalf(a(n),32)), n=0..22); # Peter Luschny, May 18 2014

CoefficientList[Series[1/(2Sqrt[1-x]) (1/Sqrt[1-x]+1/Sqrt[1-5x]),{x,0,30}],x] (* Harvey P. Dale, Mar 19 2020 *)

a(n):=sum(binomial(2*j-1,j)*binomial(n,j),j,0,n);

a(n) = Sum_{j = 0..n} binomial(2*j-1,j)*binomial(n,j).
G.f. A(x) = x*F'(x)/F(x), where F(x) is g.f. of A007317.
a(n) = T(2*n,n) for n>0, where T(n,k) is triangle of A105477.
a(n) = hypergeom([1/2,-n],[1],-4)/2 + 1/2. - Peter Luschny, May 18 2014
D-finite with recurrence: n*a(n) + (-7*n+4)*a(n-1) + (11*n-14)*a(n-2) + 5*(-n+2)*a(n-3) = 0. - R. J. Mathar, May 23 2014
2*a(n) = 1 + A026375(n). - R. J. Mathar, Jan 26 2020
From Peter Bala, Jan 09 2022: (Start)
a(n) = [x^n] ( x + 1/(1 - x) )^n.
a(0) = 1, a(1) = 2 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) - 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. (End)

A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 11, 6, 1, 1, 9, 30, 45, 30, 9, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1, 1, 18, 141, 630, 1770, 3258, 3989, 3258, 1770, 630, 141, 18, 1, 1, 21, 196, 1071, 3801, 9198, 15533, 18483, 15533, 9198, 3801, 1071, 196, 21, 1

Offset: 0

Peter Luschny, May 08 2016

From R. J. Mathar, Nov 05 2021: (Start)
These are the antidiagonals of the following array with the bivariate generating function 1/(1-x^2-3*x*y-y^2):
   0     1     0     1     0     1     0     1     0     1 ...
   3     0     6     0     9     0    12     0    15     0 ...
   0    11     0    30     0    58     0    95     0   141 ...
   6     0    45     0   144     0   330     0   630     0 ...
   0    30     0   195     0   685     0  1770     0  3801 ...
   9     0   144     0   873     0  3258     0  9198     0 ...
   0    58     0   685     0  3989     0 15533     0 46928 ...
  12     0   330     0  3258     0 18483     0 74280     0 ...
   0    95     0  1770     0 15533     0 86515     0 356283 ...
  15     0   630     0  9198     0 74280     0 408105     0 ...
   0   141     0  3801     0 46928     0 356283     0 1936881 ... (End)

			                                1;
                            1,  3, 1;
                         1, 6, 11, 6, 1;
                     1, 9, 30, 45, 30, 9, 1;
              1, 12, 58, 144, 195, 144, 58, 12, 1;
         1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 100, flattened).
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 6.
László Németh, Tetrahedron trinomial coefficient transform, Integers (2019) 19, Article A41.

Cf. A002212, A026375, A026376, A110165.

T := (n,k) -> simplify(GegenbauerC(`if`(k

Table[If[n == 0, 1, GegenbauerC[If[k < n, k, 2 n - k], -n, -3/2]], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* Michael De Vlieger, Aug 02 2019 *)

T(n,n) = A026375(n) for n>=0.
T(n,n-1) = A026376(n) for n>=1.
T(n,n+1)/n = A002212(n) for n>=1.

A276536 Binomial sums of the cubes of the central binomial coefficients.

Original entry on oeis.org

1, 9, 233, 8673, 376329, 17800209, 890215361, 46294813497, 2478150328777, 135642353562321, 7556884938829233, 427106589765940137, 24429206859151618209, 1411391470651692285609, 82245902444586364980057, 4828398428680134702936273

Offset: 0

Emanuele Munarini, Nov 16 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..554
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
The Wolfram Functions Site, Complete Elliptic Integrals, 2016.

Cf. Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^m: A026375 (m=1), A248586 (m=2), this sequence (m=3).

[&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 30 2016

Table[Sum[Binomial[n, k]Binomial[2k, k]^3, {k, 0, n}], {n, 0, 100}]

makelist(sum(binomial(n,k)*binomial(2*k,k)^3,k,0,n),n,0,12);

a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^3.
Recurrence: (n^3 + 12n^2 + 48n + 64) * a(n+4) - (68n^3 + 714n^2 + 2500n + 2919) * a(n+3) + (198n^3 + 1782n^2 + 5363n + 5397) * a(n+2) - 98 * (2n^3 + 15n^2 + 37n + 30) * a(n+1) + 65 * (n^3 + 6n^2 + 11n + 6) * a(n) = 0.
G.f.: (4/Pi^2) * K(1/2 - 1/2 * sqrt((1-65*t)/(1-t)))^2 / (1-t), where K(x) is complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
a(n) ~ 65^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016
a(n) = 4F3(1/2,1/2,1/2,-n; 1,1,1; -64). - Ilya Gutkovskiy, Nov 25 2016

A302181 Number of 3D walks of type abb.

Original entry on oeis.org

1, 5, 62, 1065, 21714, 492366, 12004740, 308559537, 8255788970, 227976044010, 6457854821340, 186814834574550, 5500292590186380, 164387681345290500, 4976887208815547640, 152378485941172462785, 4711642301137121933850, 146964278352052950118770, 4619875954522866283392300

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A005558, A126869.

C := n-> binomial(2*n, n)/(n+1): # Catalan numbers
A302181 := n-> add(binomial(2*n, k)*C(iquo(k+1, 2))*C(iquo(k, 2))*(2*iquo(k, 2)+1)*add((-1)^(k+j)*binomial(2*n-k, iquo(j,2)), j=0..2*n-k), k=0..2*n): seq(A302181(n), n = 0 .. 18); # Mélika Tebni, Nov 06 2024

a(n) = Sum_{k=0..2*n} binomial(2*n, k) * A005558(k) * A126869(2*n-k). - Mélika Tebni, Nov 06 2024

a(8)-a(18) from Nachum Dershowitz, Aug 03 2020

cp's OEIS Frontend

A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).

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

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A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

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A026380 a(n) = T(n,[ n/2 ]), where T is the array in A026374.

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A117852 Mirror image of A098473 formatted as a triangular array.

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A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

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A242586 Expansion of 1/(2*sqrt(1-x))*(1/sqrt(1-x)+1/(sqrt(1-5*x))).

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A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.

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

A242586 Expansion of 1/(2sqrt(1-x))(1/sqrt(1-x)+1/(sqrt(1-5*x))).