A073157 -id:A073157 - cp's OEIS frontend

A364339 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^6).

1, 2, 13, 150, 1978, 28603, 438273, 6992052, 114915180, 1932233883, 33081722359, 574755965137, 10107627041697, 179576579730534, 3218352405778284, 58114340679967608, 1056284029850962674, 19310039426151335622, 354818596435147647654, 6549556302551204621664, 121394125733645986376838

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Cf. A073157, A364336, A364337, A364338.

Cf. A239109, A364333, A364340.

terms = 21; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^6) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

a(n) = sum(k=0, n, binomial(6*k+1, k)*binomial(6*k+1, n-k)/(6*k+1));

a(n) = Sum_{k=0..n} binomial(6*k+1,k) * binomial(6*k+1,n-k) / (6*k+1).

A215715 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 13, 118, 1242, 14227, 172177, 2165732, 28032668, 370944717, 4995412647, 68239105203, 943278064473, 13169938895473, 185453340189492, 2630813161415976, 37561512615867450, 539336703889993006, 7783290731579783544, 112828761898680983141, 1642222504807143423470

Offset: 0

Paul D. Hanna, Aug 21 2012

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = (1 + x^r*F(x)^(p+1)) * (1 + x^(r+s)*F(x)^(p+q+1)), then
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ).
The radius of convergence of g.f. A(x) is r = 0.06368546004073732405169450... with A(r) = 1.3960637117611795281240000742797488619448782873... where y=A(r) satisfies 6*y^5 + 17*y^4 - 46*y^3 + 16*y^2 + 4*y - 8 = 0.

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 118*x^3 + 1242*x^4 + 14227*x^5 + ...
Related expansions.
A(x)^2 = 1 + 4*x + 30*x^2 + 288*x^3 + 3125*x^4 + 36490*x^5 + ...
A(x)^4 = 1 + 8*x + 76*x^2 + 816*x^3 + 9454*x^4 + 115260*x^5 + ...
A(x)^6 = 1 + 12*x + 138*x^2 + 1648*x^3 + 20427*x^4 + 260934*x^5 + ...
where A(x) = 1 + x*(A(x)^2 + A(x)^4) + x^2*A(x)^6.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x*A(x) + (1 + 2^2*A(x)^2 + A(x)^4)*x^2*A(x)^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3*A(x)^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4*A(x)^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5*A(x)^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 22*x^2/2 + 284*x^3/3 + 3878*x^4/4 + 54607*x^5/5 + 784144*x^6/6 + 11414265*x^7/7 + 167819014*x^8/8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..825
Vaclav Kotesovec, Recurrence

Cf. A198953.

Cf. A007863, A069271, A073157, A215654.

CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(1+Sqrt[1-4*x*(1+x)^2])^2/(4*(1+x)^2),{x,0,20}],x]],x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^2)*(1 + x*A^4)); polcoeff(A, n)}
for(n=0,31,print1(a(n),", "))

{a(n)=polcoeff( sqrt((1/x)*serreverse( x*(1 + sqrt(1 - 4*x*(1+x)^2 +x*O(x^n)))^2/(4*(1+x)^2))), n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*A^(2*j))*x^m*A^m/m))); polcoeff(A, n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^(2*j))*x^m*A^(3*m)/m))); polcoeff(A, n)}

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1 + sqrt(1 - 4*x*(1+x)^2))^2/(4*(1+x)^2) ) ).
(2) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(2*k) ).
The formal inverse of the g.f. A(x) is (sqrt(x^4 + 4*x^3 - 2*x^2 + 1) - (1+x^2))/(2*x^4).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 15.70217125479403872... is the root of the equation -1024 - 3840*d + 26368*d^2 - 58644*d^3 + 1933*d^4 + 108*d^5 = 0 and c = 0.320114409... - Vaclav Kotesovec, Sep 17 2013
a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(2*n+2*k+1,n-k) / (2*n+2*k+1). - Seiichi Manyama, Jul 18 2023

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261

Offset: 0

Seiichi Manyama, Jul 18 2023

Cf. A007863, A069271, A073157, A215654, A215715, A364333.
Cf. A215623, A215624, A239108, A364335, A364338.
Cf. A200719.

A364331 := proc(n)
    add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023

a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(2*n+3*k+1, n-k)/(2*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k) / (2*n+3*k+1).

x/series_reversion(x*A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + ..., the g.f. of A215623. - Peter Bala, Sep 08 2024

A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

Original entry on oeis.org

1, 0, -1, 2, -2, -1, 9, -20, 20, 24, -150, 327, -293, -599, 3097, -6452, 4854, 15878, -71252, 140112, -81328, -437346, 1746254, -3214989, 1223971, 12345295, -44552833, 76242173, -11292089, -354175849, 1167638037, -1842585992, -233903034, 10273377388, -31169512310

Offset: 0

Seiichi Manyama, Jul 20 2023

Cf. A364372, A364373, A364375.

Cf. A000108, A073157.

A364371 := proc(n)
    add((-1)^k* binomial(2*k+1,k) * binomial(2*k+1,n-k)/(2*k+1),k=0..n) ;
end proc:
seq(A364371(n),n=0..70); # R. J. Mathar, Jul 25 2023

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

G.f.: A(x) = 2*(1 + x) / (1 + sqrt(1+4*x*(1 + x)^2)).
a(n) = Sum_{k=0..n} (-1)^k * binomial(2*k+1,k) * binomial(2*k+1,n-k) / (2*k+1).
D-finite with recurrence (n+1)*a(n) +(5*n-1)*a(n-1) +6*(2*n-3)*a(n-2) +6*(2*n-5)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
A(x) = (1 + x)*c(-x*(1+x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
(1/x) * series_reversion(x/A(x)) = 1 - x^2 + 2*x^3 - 11*x^5 + 28*x^6 + ..., the g.f. of A364375. (End)

A073153 Triangle of numbers relating two sequences A073155 and A073156.

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 14, 18, 22, 36, 56, 70, 86, 100, 156, 237, 293, 349, 405, 461, 698, 1046, 1283, 1507, 1703, 1927, 2164, 3210, 4762, 5808, 6756, 7540, 8324, 9272, 10318, 15080, 22198, 26960, 31144, 34462, 37598, 40916, 45100, 49862, 72060, 105430, 127628

Offset: 0

Paul D. Hanna, Jul 29 2002

			T(4,0) = T(3,3) + 2*T(2,2) + T(1,1) = 2 + 2*9 + 36 = 56.
T(5,2) = A073155(0)*A073155(5) + A073155(1)*A073155(4) + A073155(2)*A073155(3) = 1*237 + 1*56 + 4*14 = 349.
Triangle begins:
1;
1, 2;
4, 5, 9;
14, 18, 22, 36;
56, 70, 86, 100, 156;
237, 293, 349, 405, 461, 698;
1046, 1283, 1507, 1703, 1927, 2164, 3210; ...

Cf. A073154, A073155, A073156, A073157, A073153.

Triangle {T(n, k), n >= 0, 0<=k<=n} defined by:  T(0, 0)=1,
T(n, 0) = A073155(n),  T(n, n) = A073156(n),
T(n, 0) = T(n-1, n-1) + 2*T(n-2, n-2) + T(n-3, n-3),
T(n, k) = Sum_{j=0..k} A073155(j) * A073155(n-j).

Formulas simplified by Paul D. Hanna, Nov 26 2012

a(31) onward corrected by Sean A. Irvine, Nov 18 2024

A366326 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^2).

Original entry on oeis.org

1, 2, -3, 14, -78, 479, -3131, 21372, -150588, 1087057, -7998295, 59763129, -452257495, 3459109408, -26697940390, 207672518808, -1626400971710, 12813379464399, -101482102525511, 807524595076284, -6452856224076654, 51760509258982478, -416620859045829372

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..1067

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366327, A366328.

Cf. A006013, A364393.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(3*k-1, k)*binomial(n+k-2, n-k)/(3*k-1));

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k-1,k) * binomial(n+k-2,n-k)/(3*k-1).

A073156 Main diagonal sequence of triangle A073153.

Original entry on oeis.org

1, 2, 9, 36, 156, 698, 3210, 15080, 72060, 349184, 1711869, 8475494, 42318018, 212843826, 1077391794, 5484472880, 28058940086, 144195777552, 744017466318, 3852968380624, 20019113126120, 104329129258596, 545214946753377

Offset: 0

Paul D. Hanna, Jul 29 2002

Cf. A073153, A073155, A073157.

a(n, r=2, s=2, t=2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 07 2024

Convolution of sequence A073155: a(n) = Sum_{k=0..n} A073155(k) * A073155(n-k).
G.f.: 1/4*(1-(1-4*x*(1+x)^2)^(1/2))^2/x^2/(1+x)^4. - Vladeta Jovovic, Oct 10 2003
From Seiichi Manyama, Dec 07 2024: (Start)
G.f. A(x) satisfies A(x) = ( 1 + x * (1 + x)^2 * A(x) )^2.
a(n) = Sum_{k=0..n} binomial(2*k+2,k) * binomial(2*k,n-k)/(k+1). (End)

More terms from Vladeta Jovovic, Oct 10 2003

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.

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 17, 216, 3224, 52640, 910452, 16392140, 303996224, 5767278431, 111401778266, 2183535060362, 43319505976084, 868220464851417, 17552981176788200, 357544690982030744, 7330803752675100908, 151172599088871911072, 3133367418601958989295, 65242183918761533467216

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A007863, A069271, A073157, A215654, A215715, A364331.
Cf. A239109, A364339, A364340.
Cf. A200718.

a(n) = sum(k=0, n, binomial(2*n+4*k+1, k)*binomial(2*n+4*k+1, n-k)/(2*n+4*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+4*k+1,k) * binomial(2*n+4*k+1,n-k) / (2*n+4*k+1).

A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

Original entry on oeis.org

1, 2, -1, 3, -10, 36, -137, 543, -2219, 9285, -39587, 171369, -751236, 3328218, -14878455, 67030785, -304036170, 1387247580, -6363044315, 29323149825, -135700543190, 630375241380, -2938391049395, 13739779184085, -64430797069375, 302934667061301, -1427763630578197

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.

Cf. A000108, A002212, A112478.

a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
seq(a(n), n = 1..30); # Peter Bala, Sep 10 2024

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

G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n-2,n-k)/(2*k-1).
a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2023
From Peter Bala, Sep 10 2024: (Start)
a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.
a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.
P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2)  = -1. (End)

cp's OEIS Frontend

A364339 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^6).

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A215715 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^4).

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A364331 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^5).

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A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

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A073153 Triangle of numbers relating two sequences A073155 and A073156.

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A366326 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^2).

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A073156 Main diagonal sequence of triangle A073153.

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

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A364333 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^6).

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A215715 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^4).

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

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

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).