A006139 -id:A006139 - cp's OEIS frontend

A361791 Expansion of 1/sqrt(1 - 4*x/(1+x)^5).

1, 2, -4, -10, 30, 72, -238, -580, 1970, 4910, -16734, -42750, 144600, 379000, -1264700, -3402480, 11160730, 30828070, -99168820, -281279030, 885931600, 2580541580, -7948885910, -23779051760, 71572652480, 219906488302, -646332447086, -2039738985238, 5850898295170

Offset: 0

Seiichi Manyama, Mar 24 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..2034

Cf. A006139, A137635, A360133, A361790, A361792.

Cf. A359758.

a[n_]:=(-1)^(n+1)Pochhammer[n,4]HypergeometricPFQ[{3/2,1-n,1+n/4,(5+n)/4, (6+n)/4, (7+n)/4}, {6/5,7/5,8/5,9/5,2}, 2^10/5^5]/12; Join[{1},Array[a,28]] (* Stefano Spezia, Jul 11 2024 *)

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

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k)) \\ Winston de Greef, Mar 24 2023

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+4*k-1,n-k).
n*a(n) = -( (2*n-4)*a(n-1) + (11*n-14)*a(n-2) + 20*(n-3)*a(n-3) + 15*(n-4)*a(n-4) + 6*(n-5)*a(n-5) + (n-6)*a(n-6) ) for n > 5.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+3-k,4) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,4)*hypergeom([3/2, 1-n, 1+n/4, (5+n)/4, (6+n)/4, (7+n)/4], [6/5, 7/5, 8/5, 9/5, 2], 2^10/5^5)/12 for n > 0. - Stefano Spezia, Jul 11 2024

A361792 Expansion of 1/sqrt(1 - 4*x/(1+x)^6).

Original entry on oeis.org

1, 2, -6, -10, 66, 60, -750, -236, 8682, -2098, -100792, 80286, 1162458, -1603412, -13225764, 26767020, 147428498, -409582818, -1596563202, 5941802122, 16587101544, -83014131140, -161717252990, 1126247965980, 1411774064970, -14905602076350

Offset: 0

Seiichi Manyama, Mar 24 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006139, A137635, A360133, A361790, A361791.

Cf. A360132.

a[n_]:=(-1)^(n+1)Pochhammer[n,5]HypergeometricPFQ[{1-n,1+n/5,(6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5}, {7/6,4/3,5/3,11/6,2}, 5^5/(2^4*3^6)]/60; Join[{1},Array[a,25]] (* Stefano Spezia, Jul 11 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^6))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+5*k-1,n-k).
n*a(n) = -( (3*n-5)*a(n-1) + (17*n-24)*a(n-2) + 35*(n-3)*a(n-3) + 35*(n-4)*a(n-4) + 21*(n-5)*a(n-5) + 7*(n-6)*a(n-6) + (n-7)*a(n-7) ) for n > 6.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+4-k,5) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,5)*hypergeom([1-n, 1+n/5, (6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5], [7/6, 4/3, 5/3, 11/6, 2], 5^5/(2^4*3^6))/60 for n > 0. - Stefano Spezia, Jul 11 2024

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 10, 12, 9, 1, 6, 21, 44, 63, 54, 27, 1, 8, 36, 104, 214, 312, 324, 216, 81, 1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243, 1, 12, 78, 340, 1095, 2712, 5284, 8136, 9855, 9180, 6318, 2916, 729, 1, 14, 105, 532, 2009, 5922, 13993, 26840, 41979

Offset: 0

Paul D. Hanna, Jun 01 2003

			Triangle begins:
  1;
  1,  2,  3;
  1,  4, 10,  12,   9;
  1,  6, 21,  44,  63,   54,   27;
  1,  8, 36, 104, 214,  312,  324,  216,   81;
  1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243;

Alois P. Heinz, Rows n = 0..100, flattened

Cf. A000079, A000244, A000400, A002426, A084600 - A084606, A006139, A084609 - A084615, A101822, A212697.

a084608 n = a084608_list !! n
a084608_list = concat $ iterate ([1,2,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

A084608:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*2^(k-2*j)*3^j: j in [0..k]]) >;
[A084608(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 27 2023

f:= proc(n) option remember; expand((1+2*x+3*x^2)^n) end:
T:= (n,k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

row[n_] := (1+2x+3x^2)^n + O[x]^(2n+1) // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+2*x+3*x^2)^n,k,x); print1(t",")); print(" "))

def A084608(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*2^(k-2*j)*3^j for j in range(k//2+1))
flatten([[A084608(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 27 2023

From G. C. Greubel, Mar 27 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*2^(k-2*j)*3^j.
T(n, n) = A084609(n).
T(n, 2*n-1) = A212697(n), n >= 1.
T(n, 2*n) = A000244(n).
Sum_{j=0..2*n} T(n, k) = A000400(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = A000079(n).
Sum_{k=0..n} T(n-k, k) = A101822(n). (End)

A360266 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-2k),n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 312, 1210, 4752, 18834, 75184, 301856, 1217604, 4930626, 20032052, 81615072, 333328532, 1364264250, 5594210292, 22977466864, 94517423444, 389316529512, 1605533230256, 6628467569292, 27393187077144, 113310732332274, 469101108803052

Offset: 0

Seiichi Manyama, Jan 31 2023

nonn

Diagonal of rational function 1/(1 - (x + y + x^3*y^2)). - Seiichi Manyama, Mar 23 2023

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006139, A360267.

Cf. A157004, A374598.

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

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

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

n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-3)*a(n-3).

A361812 Expansion of 1/sqrt(1 - 4x(1+x)^3).

Original entry on oeis.org

1, 2, 12, 62, 342, 1932, 11094, 64480, 378150, 2233304, 13263772, 79136844, 473969586, 2847911596, 17159547804, 103640073972, 627280131594, 3803643145596, 23102172930156, 140522319418164, 855880464524472, 5219168576004184, 31861229045809436

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A361830.

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361813, A361814.

a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a,23,0] (* Stefano Spezia, Jul 11 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^3))

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(3*k,n-k).
n*a(n) = 2 * ( (2*n-1)*a(n-1) + 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) + (2*n-4)*a(n-4) ) for n > 3.
a(n) = binomial(2*n, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 1/2-n, 2/3-n], -2^6/3^3). - Stefano Spezia, Jul 11 2024

A084606 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+2x^2)^n.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 8, 8, 4, 1, 6, 18, 32, 36, 24, 8, 1, 8, 32, 80, 136, 160, 128, 64, 16, 1, 10, 50, 160, 360, 592, 720, 640, 400, 160, 32, 1, 12, 72, 280, 780, 1632, 2624, 3264, 3120, 2240, 1152, 384, 64, 1, 14, 98, 448, 1484, 3752, 7448, 11776, 14896, 15008, 11872

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,2,2},
{1,4,8,8,4},
{1,6,18,32,36,24,8},
{1,8,32,80,136,160,128,64,16},
{1,10,50,160,360,592,720,640,400,160,32},
{1,12,72,280,780,1632,2624,3264,3120,2240,1152,384,64},

Cf. A002426, A084600-A084605, A006139, A084608-A084615.

for(n=0,15, for(k=0,2*n,t=polcoeff((1+2*x+2*x^2)^n,k,x); print1(t",")); print(" "))

A106258 Expansion of 1/sqrt(1-8x-8x^2).

Original entry on oeis.org

1, 4, 28, 208, 1624, 13024, 106336, 879232, 7338592, 61699456, 521753728, 4433024512, 37812715264, 323603221504, 2777262164992, 23893731463168, 206005885076992, 1779480850438144, 15396895523989504, 133420304211238912

Offset: 0

Paul Barry, Apr 28 2005

Central coefficient of (1+4x+6x^2)^n. Fourth binomial transform of 1/sqrt(1-24x^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)), 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).

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 4 colors and the U steps can have 6 colors. - N-E. Fahssi, Mar 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A006139, A106259, A106260, A106261.

CoefficientList[Series[1/Sqrt[1-8*x-8*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
RecurrenceTable[{a[0]==1,a[1]==4,a[n]==(4(2n-1)a[n-1]+8(n-1)a[n-2])/n}, a,{n,20}] (* Harvey P. Dale, Mar 13 2013 *)

E.g.f.: exp(4*x)*BesselI(0, 4*sqrt(3/2)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)2^k}.
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(18+6*sqrt(6))*(4+2*sqrt(6))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012

A106259 Expansion of 1/sqrt(1-12x-12x^2).

Original entry on oeis.org

1, 6, 60, 648, 7344, 85536, 1014336, 12182400, 147702528, 1803907584, 22159733760, 273508669440, 3389106769920, 42134712606720, 525323149885440, 6565657319866368, 82235651779657728, 1031956779869798400

Offset: 0

Paul Barry, Apr 28 2005

Central coefficient of (1+6x+12x^2)^n. Sixth binomial transform of 1/sqrt(1-48x^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)), 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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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, A106260, A106261.

CoefficientList[Series[1/Sqrt[1-12*x-12*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

E.g.f.: exp(6*x)*BesselI(0, 6*sqrt(4/3)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)3^k}.
D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) + 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ (1+sqrt(3))*(6+4*sqrt(3))^n/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 17 2012

A106260 Expansion of 1/sqrt(1-16x-16x^2).

Original entry on oeis.org

1, 8, 104, 1472, 21856, 333568, 5183744, 81590272, 1296426496, 20750839808, 334081306624, 5404163080192, 87763693060096, 1430025994108928, 23367175920287744, 382767375745810432, 6283401962864377856

Offset: 0

Paul Barry, Apr 28 2005

Central coefficient of (1+8x+20x^2)^n. Eighth binomial transform of 1/sqrt(1-80x^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)), 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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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, A106261.

CoefficientList[Series[1/Sqrt[1-16*x-16*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

E.g.f.: exp(8*x)*BesselI(0, 8*sqrt(5/4)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)4^k}.
D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) + 16*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(50+20*sqrt(5))*(8+4*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012

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

Original entry on oeis.org

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

cp's OEIS Frontend

A361791 Expansion of 1/sqrt(1 - 4*x/(1+x)^5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A361792 Expansion of 1/sqrt(1 - 4*x/(1+x)^6).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+3*x^2)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

SageMath

Formula

A360266 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-2*k),n-2*k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A361812 Expansion of 1/sqrt(1 - 4*x*(1+x)^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A084606 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+2x^2)^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A106258 Expansion of 1/sqrt(1-8x-8x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A106259 Expansion of 1/sqrt(1-12x-12x^2).

Views

Author

Keywords

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

A360266 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-2k),n-2*k).

A361812 Expansion of 1/sqrt(1 - 4x(1+x)^3).

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