A006139 -id:A006139 - cp's OEIS frontend

A106261 Expansion of 1/sqrt(1 - 20x - 20x^2).

1, 10, 160, 2800, 51400, 970000, 18640000, 362800000, 7128700000, 141103000000, 2809273600000, 56197096000000, 1128614356000000, 22741607080000000, 459548117440000000, 9309106936000000000, 188980474087000000000

Offset: 0

Paul Barry, Apr 28 2005

Central coefficient of (1 + 10x + 30x^2)^n. Tenth binomial transform of 1/sqrt(1 - 120x^2). In general, 1/sqrt(1 - 4*r*x - 4*r*x^2) has e.g.f. exp(2rx)*BesselI(0,2r*sqrt((r+1)/r)x)), and a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k gives the central coefficient of (1 + (2r)*x + r(r+1)*x^2) and is the (2r)-th binomial transform of 1/sqrt(1 - 8*C(n+1,2)x^2).

G. C. Greubel, Table of n, a(n) for n = 0..750
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Cf. A006139, A106258, A106259, A106260.

CoefficientList[Series[1/Sqrt[1-20*x-20*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2013 *)

for(n=0,25, print1(sum(k=0,n,binomial(2*k,k)*binomial(k,n-k)*5^k), ", ")) \\ G. C. Greubel, Jan 31 2017

E.g.f.: exp(10*x)*BesselI(0, 10*sqrt(6/5)*x).
a(n) = Sum_{k=0..n} C(2k, k)*C(k, n-k)*5^k.
D-finite with recurrence: n*a(n) + 10*(-2*n+1)*a(n-1) + 20*(-n+1)*a(n-2) = 0. - R. J. Mathar, Nov 26 2012
a(n) ~ sqrt((1+sqrt(5/6))/2) * (10+2*sqrt(30))^n / sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2013

A115951 Expansion of 1/sqrt(1-4xy-4x^2y).

Original entry on oeis.org

1, 0, 2, 0, 2, 6, 0, 0, 12, 20, 0, 0, 6, 60, 70, 0, 0, 0, 60, 280, 252, 0, 0, 0, 20, 420, 1260, 924, 0, 0, 0, 0, 280, 2520, 5544, 3432, 0, 0, 0, 0, 70, 2520, 13860, 24024, 12870, 0, 0, 0, 0, 0, 1260, 18480, 72072, 102960, 48620, 0, 0, 0, 0, 0, 252, 13860, 120120, 360360, 437580, 184756

Offset: 0

Paul Barry, Mar 14 2006

Row sums are A006139. Diagonal sums are A115962.

Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - Ralf Stephan, Apr 07 2016.

			Triangle begins
   1,
   0,  2,
   0,  2,  6,
   0,  0, 12,  20,
   0,  0,  6,  60,  70,
   0,  0,  0,  60, 280,  252,
   0,  0,  0,  20, 420, 1260, 924

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).

/* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 02 2015 *)

{T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ G. C. Greubel, May 06 2019

[[binomial(2*k,k)*binomial(k,n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2019

Number triangle T(n,k) = C(2k,k)*C(k,n-k).
From Peter Bala, Sep 02 2015: (Start)
Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.
P * M * P^-1 is a signed version of A063007. (End)

A374497 Expansion of 1/(1 - 4x - 4x^2)^(3/2).

Original entry on oeis.org

1, 6, 36, 200, 1080, 5712, 29792, 153792, 787680, 4009280, 20304768, 102405888, 514678528, 2579028480, 12890311680, 64283809792, 319954540032, 1589720712192, 7886437652480, 39069462835200, 193307835764736, 955361266917376, 4716674314223616, 23264437702656000

Offset: 0

Seiichi Manyama, Jul 09 2024

nonn

Cf. A001788, A002457, A025163.
Cf. A006139, A374511, A374513.
Cf. A071356.

a[n_]:= Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-k],{k,0,n}]; Array[a,24,0] (* Stefano Spezia, May 08 2025 *)

a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

a(0) = 1, a(1) = 6; a(n) = (2*(2*n+1)*a(n-1) + 4*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A071356(n).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 20 2025

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4kx - 4*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 4, 8, 0, 1, 6, 26, 32, 6, 1, 8, 56, 184, 136, 0, 1, 10, 98, 576, 1366, 592, 20, 1, 12, 152, 1328, 6216, 10424, 2624, 0, 1, 14, 218, 2560, 18886, 68976, 80996, 11776, 70, 1, 16, 296, 4392, 45256, 276208, 779456, 637424, 53344, 0

Offset: 0

Seiichi Manyama, Aug 29 2025

			Square array begins:
   1,    1,     1,      1,       1,        1,        1, ...
   0,    2,     4,      6,       8,       10,       12, ...
   2,    8,    26,     56,      98,      152,      218, ...
   0,   32,   184,    576,    1328,     2560,     4392, ...
   6,  136,  1366,   6216,   18886,    45256,    92886, ...
   0,  592, 10424,  68976,  276208,   822800,  2020392, ...
  20, 2624, 80996, 779456, 4114004, 15235520, 44758244, ...

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

Columns k=0..3 give A126869, A006139, A098443, A387428.

Main diagonal gives A387430.

a(n, k) = sum(j=0, n\2, (k^2+1)^j*(2*k)^(n-2*j)*binomial(n, 2*j)*binomial(2*j, j));

A(n,k) = Sum_{j=0..n} (k-i)^j * (k+i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*k*(2*n-1)*A(n-1,k) + 4*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * (2*k)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*k*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*k*x) * BesselI(0, 2*sqrt(k^2+1)*x).

A179191 Expansion of o.g.f. (1/2)(-1 + 1/sqrt(1 - 4x - 4*x^2)).

Original entry on oeis.org

0, 1, 4, 16, 68, 296, 1312, 5888, 26672, 121696, 558464, 2574848, 11917952, 55345408, 257741824, 1203224576, 5629027072, 26383656448, 123868321792, 582414688256, 2742116907008, 12926036258816, 60998951747584, 288147689046016, 1362407763795968, 6447125560016896

Offset: 0

Clark Kimberling, Jul 01 2010

nonn

G.f. A(x) satisfies A(x)^2 + A(x) = (x^2 + x)/(1 - 4*x - 4*x^2). - Michael Somos, Jan 28 2019

			G.f. = x + 4*x^2 + 16*x^3 + 68*x^4 + 296*x^5 + 1312*x^6 + 5888*x^7 + ....

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.

Cf. A178693, A178694, A179190.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (-1 + 1/Sqrt(1-4*x-4*x^2))/2 )); // G. C. Greubel, Jan 25 2019

CoefficientList[1/2 (-1 + (1-4x-4x^2)^(-1/2)) + O[x]^23, x] (* Jean-François Alcover, Jul 27 2018 *)

a(n):=sum(m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n),m,1,n); /* Vladimir Kruchinin, Mar 11 2011 */
a(n):=sum(2^(n-k-1)*binomial(n,k)*binomial(n-k,k),k,0,n); /* Vladimir Kruchinin, Mar 12 2015 */

my(x='x+O('x^30)); concat([0], Vec((-1 +1/sqrt(1-4*x-4*x^2))/2)) \\ G. C. Greubel, Jan 25 2019

((-1 + 1/sqrt(1-4*x-4*x^2))/2).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 25 2019

G.f.: (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).
G.f.: A(x) = x*A001006(A000045(x)/x-1)/(1-x*A001006(A000045(x)/x-1)).
a(n) = Sum_{m=1..n} m*Sum_{k=m..n} (Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i))*(Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j))/k. - Vladimir Kruchinin, Mar 11 2011
a(n) = Sum_{k=0..n} 2^(n-k-1)*binomial(n,k)*binomial(n-k,k). - Vladimir Kruchinin, Mar 12 2015
From Vaclav Kotesovec, Jan 26 2019: (Start)
D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) + 4*(n-1)*a(n-2).
a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). (End)
0 = a(n)*(16*a(n+1) +24*a(n+2) -8*a(n+3)) + a(n+1)*(+8*a(n+1) +16*a(n+2) -6*a(n+3)) + a(n+2)*(-2*a(n+2) +a(n+3)) for all n in Z except n=-1. - Michael Somos, Jan 27 2019
a(n) = A006139(n)/2, n>0. - R. J. Mathar, Jan 24 2020

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

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

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

Original entry on oeis.org

1, 2, 14, 80, 486, 3030, 19184, 122924, 794678, 5173160, 33863666, 222683588, 1469908848, 9733916596, 64636957300, 430240178484, 2869778018070, 19177245746844, 128361805431752, 860443079597872, 5775392952659170, 38811408514848032, 261101034656317244

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361814.

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

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) + 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

Original entry on oeis.org

1, 2, 16, 100, 660, 4482, 30886, 215364, 1515000, 10730800, 76426846, 546792056, 3926775646, 28290272420, 204375145480, 1479963148220, 10739326203132, 78072933869364, 568503202324540, 4145718464390120, 30271771382355430, 221305746414518180

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1139

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361813.

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

a(n)= sum(k=0, n, binomial(2*k,k) * binomial(5*k,n-k)) \\ Winston de Greef, Mar 25 2023

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 5*(2*n-2)*a(n-2) + 10*(2*n-3)*a(n-3) + 10*(2*n-4)*a(n-4) + 5*(2*n-5)*a(n-5) + (2*n-6)*a(n-6) ) for n > 5.

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    2,   2,    2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   46,   62,   80,  100, ...
   70, 136,  226,  342,  486,  660, ...
  252, 592, 1136, 1932, 3030, 4482, ...

Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Cf. A099233, A361834.

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

G.f. of column k: 1/sqrt(1 - 4*x*(1+x)^k).

n*T(n,k) = 2 * Sum_{j=0..k} binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A374511 Expansion of 1/(1 - 4x - 4x^2)^(5/2).

Original entry on oeis.org

1, 10, 80, 560, 3640, 22512, 134400, 781440, 4451040, 24939200, 137865728, 753625600, 4080643840, 21916106240, 116877312000, 619457482752, 3265293719040, 17128725519360, 89462514606080, 465434423336960, 2412895587536896, 12468681310412800, 64242981906022400

Offset: 0

Seiichi Manyama, Jul 09 2024

nonn

Cf. A006139, A374497, A374513.

a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a,23,0] (* Stefano Spezia, Jul 10 2024 *)

a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));

a(0) = 1, a(1) = 10; a(n) = (2*(2*n+3)*a(n-1) + 4*(n+3)*a(n-2))/n.
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = 2^(n-3)*Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], 2)/3. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

cp's OEIS Frontend

A106261 Expansion of 1/sqrt(1 - 20*x - 20*x^2).

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A115951 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).

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A374497 Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).

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A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*k*x - 4*x^2).

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A179191 Expansion of o.g.f. (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).

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A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*k*x + k*(k-4)*x^2).

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A361813 Expansion of 1/sqrt(1 - 4*x*(1+x)^4).

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A361814 Expansion of 1/sqrt(1 - 4*x*(1+x)^5).

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A106261 Expansion of 1/sqrt(1 - 20x - 20x^2).

A115951 Expansion of 1/sqrt(1-4xy-4x^2y).

A374497 Expansion of 1/(1 - 4x - 4x^2)^(3/2).

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4kx - 4*x^2).

A179191 Expansion of o.g.f. (1/2)(-1 + 1/sqrt(1 - 4x - 4*x^2)).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

A374511 Expansion of 1/(1 - 4x - 4x^2)^(5/2).