A186997 -id:A186997 - cp's OEIS frontend

A240722 Expansion of log'((-1-sqrt(1-8x)+sqrt(2+2sqrt(1-8x)+8x))/(4*x)).

1, 7, 46, 323, 2326, 17062, 126764, 950819, 7184422, 54604082, 416990564, 3197008718, 24592858876, 189719297068, 1467180979096, 11370769197347, 88291427756294, 686715981892666, 5349188181210548, 41723881423351898, 325845079538136596

Offset: 0

Vladimir Kruchinin, Apr 11 2014

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

Cf. A186997.

[(n+1)*(&+[Binomial(k+1,n-k)*Binomial(n+2*k+2, n+k+1)/(n+k+2): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 30 2018

CoefficientList[Series[(1/(16*x*(1 + x)*(-1 + 8*x)))*(8 - Sqrt[2]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x] - 3*Sqrt[2 - 16*x]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x] + 8*x*(-7 - 8*x + Sqrt[2]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 12 2014 *)

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

for(n=0, 30, print1((n+1)*sum(k=0, n, (binomial(k+1, n-k)*binomial(n +2*k +2, n+k+1))/(n+k+2)), ", ")) \\ G. C. Greubel, Sep 30 2018

a(n) = (n+1)*Sum_{k=0..n} (binomial(k+1, n-k) * binomial(n+2*k+2, n+k+1) / (n+k+2)).
A(x) = B'(x)/B(x) where B(x) = (-1-sqrt(1-8*x) + sqrt(2+2*sqrt(1-8*x)+8*x)) / (4*x) is the g.f. of A186997.
a(n) ~ 2^(3*n+2)/sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 12 2014
Conjecture: -(3*n-1)*(2*n+1)*(n+1)*a(n) +2*(21*n^3+14*n^2-2*n-1)*a(n-1) +8*n*(3*n+2)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
From Peter Bala, Jul 27 2024: (Start)
The following remarks assume an offset of 1.
a(n) = [x^n] c(x + x^2)^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Calculation suggests that the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) hold for all primes p >= 3. (End)

A243034 Expansion of A(x) = xF'(x)/(F(x) - F(x)^2), where F(x) = (-1 - sqrt(1-8x) + sqrt(2 + 2sqrt(1-8x) + 8*x))/4.

Original entry on oeis.org

1, 2, 10, 62, 422, 2992, 21736, 160442, 1197798, 9018656, 68355820, 520851212, 3986036204, 30615867128, 235879185188, 1822138940482, 14108173076358, 109454660444336, 850687921793836, 6622072711690452, 51621868156476212, 402929115540626240, 3148664886787313728

Offset: 0

Vladimir Kruchinin, May 29 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A186997.

Table[1+n*Sum[Sum[Binomial[k, n-m-k]*Binomial[n+2*k-1, n+k-1]/(n+k), {k, 1, n-m}], {m, 0, n}],{n,0,20}] (* Vaclav Kotesovec, May 31 2014 after Vladimir Kruchinin *)

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

for(n=0,25, print1(1 + n*sum(m=0,n, sum(k=1,n-m, (binomial(k,n-m-k)*binomial(n+2*k-1,n+k-1))/(n+k))), ", ")) \\ G. C. Greubel, Jun 01 2017

a(n) = 1 + n*Sum_{m=0..n} Sum_{k=1..(n-m)} binomial(k, n-m-k) * binomial(n+2*k-1, n+k-1) / (n+k).
G.f.: A(x) = x*F'(x)/(F(x)-F(x)^2), where F(x)/x is g.f. of A186997.
a(n) ~ (3+5*sqrt(3)) * 8^n / (33*sqrt(Pi*n)). - Vaclav Kotesovec, May 31 2014

A243163 Triangle read by rows: T(n,k) = binomial(k,n-k)binomial(n+2k,n+k) /(n+k+1), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 7, 12, 0, 0, 4, 45, 55, 0, 0, 0, 55, 286, 273, 0, 0, 0, 22, 546, 1820, 1428, 0, 0, 0, 0, 455, 4760, 11628, 7752, 0, 0, 0, 0, 140, 6120, 38760, 74613, 43263, 0, 0, 0, 0, 0, 3876, 67830, 302841, 480700, 246675, 0, 0, 0, 0, 0, 969, 65835

Offset: 0

Peter Luschny, May 31 2014

			                               1;
                              0, 1;
                            0, 1, 3;
                          0, 0, 7, 12;
                        0, 0, 4, 45, 55;
                     0, 0, 0, 55, 286, 273;

Cf. A001764, A002293, A186997.

T := (n, k) -> binomial(k,n-k)*binomial(n+2*k,n+k)/(n+k+1);
seq(print(seq(T(n,k), k=0..n)), n=0..10);

T(n, n) = A001764(n).
T(2*n,n) = A002293(n).
Sum_{k=0..n} T(n, k) = A186997(n).

cp's OEIS Frontend

A240722 Expansion of log'((-1-sqrt(1-8x)+sqrt(2+2sqrt(1-8x)+8x))/(4*x)).

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A243034 Expansion of A(x) = xF'(x)/(F(x) - F(x)^2), where F(x) = (-1 - sqrt(1-8x) + sqrt(2 + 2sqrt(1-8x) + 8*x))/4.

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A243163 Triangle read by rows: T(n,k) = binomial(k,n-k)binomial(n+2k,n+k) /(n+k+1), n>=0, 0<=k<=n.

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cp's OEIS Frontend

A240722 Expansion of log'((-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/(4*x)).

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Magma

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A243034 Expansion of A(x) = x*F'(x)/(F(x) - F(x)^2), where F(x) = (-1 - sqrt(1-8*x) + sqrt(2 + 2*sqrt(1-8*x) + 8*x))/4.

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Mathematica

Maxima

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Formula

A243163 Triangle read by rows: T(n,k) = binomial(k,n-k)*binomial(n+2*k,n+k) /(n+k+1), n>=0, 0<=k<=n.

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A240722 Expansion of log'((-1-sqrt(1-8x)+sqrt(2+2sqrt(1-8x)+8x))/(4*x)).

A243034 Expansion of A(x) = xF'(x)/(F(x) - F(x)^2), where F(x) = (-1 - sqrt(1-8x) + sqrt(2 + 2sqrt(1-8x) + 8*x))/4.

A243163 Triangle read by rows: T(n,k) = binomial(k,n-k)binomial(n+2k,n+k) /(n+k+1), n>=0, 0<=k<=n.