A137635 -id:A137635 - cp's OEIS frontend

A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.

1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, 1840680, 9792986, 52296799, 280163091, 1504969409, 8103433329, 43722788132, 236340999038, 1279602656590, 6938126362948, 37668424608552, 204751452911832, 1114151447523038

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

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

Cf. A137634, A137635, A137637, A137638, A073157.

{a(n)=sum(k=0,n,binomial(2*k+1,k)*binomial(2*k+1,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G,n,x)}

G.f.: A(x) = R(x)*G(x), where R(x) = 1/sqrt(1-4x(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4x(1+x)^2))/(2x(1+x)) is the g.f. of A073157.
D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-6*n-1)*a(n-2) +2*(-6*n+1)*a(n-3) +2*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Jun 23 2023
a(n) ~ sqrt((172 + (86*(78905 - 519*sqrt(129)))^(1/3) + (86*(78905 + 519*sqrt(129)))^(1/3))/129) * ((4 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 25 2023

A137637 a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.

Original entry on oeis.org

1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, 3868024, 20785035, 111931154, 603938905, 3264309644, 17671408012, 95800342628, 520022296700, 2826089180652, 15374990077568, 83727902852188, 456370687687082

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

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

Cf. A137634, A137635, A137636, A137638, A073157.

{a(n)=sum(k=0,n,binomial(2*k+2,k)*binomial(2*k+2,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G^2,n,x)}

G.f.: A(x) = R(x)*G(x)^2, where R(x) = 1/sqrt(1-4*x*(1+x)^2) is the g.f. of A137635 and G(x) = (1-sqrt(1-4*x*(1+x)^2))/(2*x*(1+x)) is the g.f. of A073157.

A360133 Expansion of 1/sqrt(1 - 4*x/(1+x)^3).

Original entry on oeis.org

1, 2, 0, -4, -4, 6, 18, 4, -48, -70, 60, 288, 170, -686, -1386, 432, 4928, 4806, -9684, -27572, -3672, 84106, 118162, -122388, -537834, -284830, 1386840, 2688944, -1103362, -10181934, -9354198, 21404728, 57921144, 3663942, -185437360, -248708676, 292137656

Offset: 0

Seiichi Manyama, Mar 24 2023

sign

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

Cf. A006139, A137635, A361790, A361791, A361792.

a[n_]:=(-1)^(n+1)n(n+1)HypergeometricPFQ[{3/2,1-n,1+n/2,(3+n)/2}, {4/3,5/3,2}, 2^4/3^3]; Join[{1},Array[a,36]] (* Stefano Spezia, Jul 11 2024 *)

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^3))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+2*k-1,n-k).
n*a(n) = -( -2*a(n-1) + (2*n)*a(n-2) + 4*(n-3)*a(n-3) + (n-4)*a(n-4) ) for n > 3.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+1-k,2) * a(k).
a(n) = (-1)^(n+1)*n*(n + 1)*hypergeom([3/2, 1-n, 1+n/2, (3+n)/2], [4/3, 5/3, 2], 2^4/3^3) for n > 0. - Stefano Spezia, Jul 11 2024

A361790 Expansion of 1/sqrt(1 - 4*x/(1+x)^4).

Original entry on oeis.org

1, 2, -2, -8, 6, 42, -8, -228, -90, 1210, 1238, -6116, -10864, 28574, 80932, -116248, -548010, 339678, 3455686, 173208, -20452674, -14036418, 113365140, 156407916, -580805472, -1312098918, 2659610562, 9621079540, -9902139124, -64566648122, 18521111032

Offset: 0

Seiichi Manyama, Mar 24 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..2497

Cf. A006139, A137635, A360133, A361791, A361792.

a[n_]:=(-1)^(n+1)Pochhammer[n,3]HypergeometricPFQ[{1-n,1+n/3,(4+n)/3, (5+n)/3}, {5/4,7/4,2}, 3^3/2^6]/3; Join[{1},Array[a,30]] (* Stefano Spezia, Jul 11 2024 *)

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^4))

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

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

Original entry on oeis.org

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

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

A361815 Expansion of 1/sqrt(1 - 4x(1-x)^2).

Original entry on oeis.org

1, 2, 2, -2, -14, -32, -30, 64, 346, 752, 584, -2044, -9486, -19324, -11368, 66180, 271658, 514916, 192584, -2151612, -7949736, -13933280, -1779028, 69933368, 235295106, 378579404, -61171228, -2267724644, -7003832456, -10248117752, 5236354188, 73288104568

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Diagonal of rational function 1/(1 - (1 - x*y) * (x + y)).

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

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361816, A361817.

Cf. A137635.

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^2))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(2*k,n-k).

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

A137634 Square array where T(n,k) = Sum_{j=0..k} C(n+2j,j)C(n+2*j,k-j), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 4, 10, 1, 6, 19, 46, 1, 8, 32, 94, 226, 1, 10, 49, 170, 474, 1136, 1, 12, 70, 282, 899, 2431, 5810, 1, 14, 95, 438, 1577, 4764, 12609, 30080, 1, 16, 124, 646, 2600, 8701, 25318, 65972, 157162, 1, 18, 157, 914, 4076, 15000, 47682, 134964, 347524, 826992

Offset: 0

Paul D. Hanna, Jan 31 2008

			Square array begins:
1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, ...;
1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, ...;
1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, ...;
1, 8, 49, 282, 1577, 8701, 47682, 260384, 1419436, ...;
1, 10, 70, 438, 2600, 15000, 85102, 477808, 2664539, ...;
1, 12, 95, 646, 4076, 24643, 145099, 839620, 4800849, ...;
1, 14, 124, 914, 6129, 38868, 237842, 1420660, 8342297, ...;
1, 16, 157, 1250, 8899, 59201, 376740, 2325088, 14036647, ...; ...

Cf. A137635, A137636, A137637, A137638, A073157.

{T(n,k)=sum(j=0,k,binomial(n+2*j,j)*binomial(n+2*j,k-j))} /* Using the g.f.: */ {T(n,k)=local(Oy=y*O(y^(n+k))); polcoeff(polcoeff(1/sqrt(1-4*y*(1+y)^2+Oy)* 1/(1-x*((1-sqrt(1-4*y*(1+y)^2+Oy))/(2*y*(1 + y+Oy))+x*O(x^n))),n,x),k,y)}

G.f.: A(x,y) = R(y)/(1 - x*G(y)), so that the g.f. of row n = R(y)*G(y)^n, where R(y) = 1/sqrt(1-4*y*(1+y)^2) and G(y) = (1-sqrt(1-4*y*(1+y)^2))/(2*y*(1+y)) is the g.f. of A073157.

A137638 Antidiagonal sums of square array A137634.

Original entry on oeis.org

1, 3, 15, 72, 361, 1840, 9505, 49578, 260540, 1377328, 7316373, 39020372, 208809544, 1120621368, 6029023185, 32507001876, 175604614108, 950233307930, 5149691511432, 27946158749572, 151843410356906, 825949622559366

Offset: 0

Paul D. Hanna, Jan 31 2008

nonn

Cf. A137634, A137635, A137636, A137637.

{a(n)=sum(k=0,n,sum(j=0,k,binomial(2*j+n-k,j)*binomial(2*j+n-k,k-j)))} /* Using the g.f.: */ {a(n)=local(G=sqrt(1 - 4*x*(1+x)^2 +x*O(x^n))); polcoeff(2*(1+x)/((1+2*x+G)*G),n)}

G.f.: A(x) = 2*(1+x)/((1+2*x + G(x))*G(x)) where G(x) = sqrt(1 - 4*x*(1+x)^2).
a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n-k+2*j,j)*C(n-k+2*j,k-j).
D-finite with recurrence 2*(n+1)*a(n) +(-3*n-7)*a(n-1) +2*(-17*n+10)*a(n-2) +8*(-7*n+10)*a(n-3) +2*(-18*n+37)*a(n-4) +4*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Jun 23 2023

cp's OEIS Frontend

A137636 a(n) = Sum_{k=0..n} C(2k+1,k)*C(2k+1,n-k) ; equals row 1 of square array A137634; also equals the convolution of A137635 and A073157.

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A137637 a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.

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A360133 Expansion of 1/sqrt(1 - 4*x/(1+x)^3).

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A361790 Expansion of 1/sqrt(1 - 4*x/(1+x)^4).

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

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

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

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A361815 Expansion of 1/sqrt(1 - 4*x*(1-x)^2).

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

A361815 Expansion of 1/sqrt(1 - 4x(1-x)^2).

A137634 Square array where T(n,k) = Sum_{j=0..k} C(n+2j,j)C(n+2*j,k-j), read by antidiagonals.