A064613 -id:A064613 - cp's OEIS frontend

A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.

1, 2, 7, 46, 485, 7066, 130987, 2946182, 77923561, 2369742130, 81467904431, 3124302688222, 132237820201357, 6123150708289226, 307903794151741075, 16709463201832993846, 973385368533058021457, 60583668821975488285282, 4012342371757905842648791, 281735471040327667890013070

Offset: 0

Vaclav Kotesovec, Nov 23 2021

nonn

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 2, 1, 0, 1, 2, 2, 1, 0, ...] with period 5. In particular, a(5*n+4) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A007317, A064613, A292632, A349603, A349639, A224500.

gf := exp(x)*(1 - sqrt(1 - 4*x))/(2*x): ser := series(gf, x, 24):
seq(n!*coeff(ser, x, n), n = 0..19);
# Alternative:
a := n -> `if`(n < 4, [1, 2, 7, 46][n + 1], ((4*n^2 - 12*n + 8)*a(n - 3) - (8*n^2 - 13*n + 5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1)):
seq(a(n), n = 0..19);  # Peter Luschny, Nov 23 2021
# Alternative
seq(simplify(hypergeom([-n, 1/2, 1], [2], -4)), n = 0..19); # Peter Bala, Mar 13 2025

Table[Sum[Binomial[n, j]*CatalanNumber[j]*j!, {j, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k!); \\ Michel Marcus, Nov 23 2021

a(n) ~ 2^(2*n + 1/2) * n^(n-1) / exp(n - 1/4).
From Peter Luschny, Nov 23 2021: (Start)
a(n) = n! * [x^n](exp(x)*(1 - sqrt(1 - 4*x))/(2*x)).
a(n) = (4*(n-1)*(n-2)*a(n - 3) - (n-1)*(8*n-5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1) for n >= 4.
a(n-1) = A224500(n) / n for n >= 1. (End)
a(n) = hypergeom([-n, 1/2, 1], [2], -4). - Peter Bala, Mar 13 2025

A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 17, 21, 54, 74, 183, 272, 644, 1025, 2342, 3928, 8734, 15264, 33227, 59989, 128484, 238008, 503563, 952038, 1995955, 3835381, 7987092, 15548654, 32223061, 63388488, 130918071, 259724317, 535168956, 1069025128

Offset: 0

José Luis Ramírez Ramírez, Apr 21 2015

nonn

			For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Robert Israel, Table of n, a(n) for n = 0..3087

Cf. A000108, A090344, A086622, A257178, A007317, A064613, A104455, A104498.

f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2},a(n),remember):
map(f, [$0..100]); # Robert Israel, Nov 04 2019

a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2,k) )*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2))),k,0,(n)/3); /* Vladimir Kruchinin, Apr 02 2016 */

G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).
a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - Vladimir Kruchinin, Apr 02 2016
(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - Robert Israel, Nov 04 2019

A271025 A(n, k) is the n-th binomial transform of the Catalan sequence (A000108) evaluated at k. Array read by descending antidiagonals for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 15, 10, 4, 1, 42, 51, 37, 17, 5, 1, 132, 188, 150, 77, 26, 6, 1, 429, 731, 654, 371, 141, 37, 7, 1, 1430, 2950, 3012, 1890, 798, 235, 50, 8, 1, 4862, 12235, 14445, 10095, 4706, 1539, 365, 65, 9, 1, 16796, 51822, 71398, 56040, 28820, 10392, 2726, 537, 82, 10, 1

Offset: 0

John M. Campbell, Mar 28 2016

Interestingly, the determinant of the n X n array of entries of the form A(i,j) is equal to the (n-1)-th superfactorial number (see A000178).
As indicated in A104455, the k-th binomial transform of A000108 will have:
   o.g.f.: (1-sqrt((1-(k+4)*x)/(1-k*x)))/(2*x),
   e.g.f.: exp((k+2)*x)*(BesselI(0,2x) - BesselI(1,2x)) and
   a(n) = Sum_{i=0..n} binomial(n, i)*CatalanNumber(i)*k^(n-i).
The columns of this array are polynomial integer sequences. The successive polynomials corresponding to the columns of this array are: p0(n) = 1, p1(n) = n + 1, p2(n) = n^2 + 2n + 2, p3(n) = n^3 + 3*n^2 + 6*n + 5, p4(n) = n^4 + 4*n^3 + 12*n^2 + 20*n + 14, and so forth. The coefficients of these successive polynomials form a number triangle, which is given by A098474.

			The array given by integers of the form A(n,k) is illustrated below:
[0] 1, 1,  2,   5,    14,    42,     132,     429,      1430, ...
[1] 1, 2,  5,   15,   51,    188,    731,     2950,     12235, ...
[2] 1, 3,  10,  37,   150,   654,    3012,    14445,    71398, ...
[3] 1, 4,  17,  77,   371,   1890,   10095,   56040,    320795, ...
[4] 1, 5,  26,  141,  798,   4706,   28820,   182461,   1188406, ...
[5] 1, 6,  37,  235,  1539,  10392,  72267,   516474,   3783115, ...
[6] 1, 7,  50,  365,  2726,  20838,  162996,  1303485,  10642310, ...
[7] 1, 8,  65,  537,  4515,  38654,  337007,  2991340,  27013723, ...
[8] 1, 9,  82,  757,  7086,  67290,  648420,  6340365,  62893270, ...
[9] 1, 10, 101, 1031, 10643, 111156, 1174875, 12568686, 136080971, ...
Seen as a triangle:
                          1
                         1, 1
                       2, 2, 1
                      5, 5, 3, 1
                   14, 15, 10, 4, 1
                 42, 51, 37, 17, 5, 1
             132, 188, 150, 77, 26, 6, 1
          429, 731, 654, 371, 141, 37, 7, 1
      1430, 2950, 3012, 1890, 798, 235, 50, 8, 1

Cf. A098474, A000178. Rows 0-5: A000108, A007317, A064613, A104455, A104498, A154623. Columns 0-3: A000012, A000027, A002522, A005491.

A := (n, k) -> (2/Pi)*int((k+4*x^2)^(n-k)*sqrt(1 - x^2), x=-1..1):
for n from 0 to 9 do seq(A(n,k), k=0..n) od; # Peter Luschny, Jan 27 2020

A000108[n_]:= Binomial[2*n,n]/(n+1) ;
T[i_,j_]: Sum[Binomial(j,k)*A000108(k)*i^(j-k), {k,0,j}] ;
A[0, k_] := CatalanNumber[k]; A[n_, k_] := n^k*Hypergeometric2F1[1/2, -k, 2, -4/n];
Table[A[n, k], {n, 0, 6}, {k, 0, 8}] (* Peter Luschny, Jan 27 2020 *)

def A000108(n): return binomial(2*n,n)/(n+1) ;
def T(i,j): return sum(binomial(j,k)*A000108(k)*i^(j-k) for k in range(j+1))

A(0,j) = A000108(j).
A(i,j) = Sum_{k=0..j} binomial(j,k)*A(i-1,k) for i >= 1.
A(i,j) = Sum_{k=0..j} binomial(j,k)*A000108(k)*i^(j-k).
From Peter Luschny, Jan 27 2020: (Start)
A(n,k) = n^k*hypergeom([1/2, -k], [2], -4/n) for n >= 1.
A(n,k) = (2/Pi)*Integral_{x=-1..1}(k + 4*x^2)^(n - k)*sqrt(1 - x^2). (End)

A171515 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033543.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 16, 14, 6, 1, 62, 52, 27, 8, 1, 270, 213, 116, 44, 10, 1, 1257, 948, 513, 216, 65, 12, 1, 6096, 4470, 2376, 1038, 360, 90, 14, 1, 30398, 21904, 11468, 5056, 1880, 556, 119, 16, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to B^2*A039598*B^(-2), B = A007318.

			Triangle begins : 1 ; 2,1 : 5,4,1 ; 16,14,6,1 ; 62,52,27,8,1 ; ...

Cf. A052179, A171486

Sum_{k, 0<=k<=n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = 0, 1, 2, 3 respectively.

T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*2^i. - Philippe Deléham, Feb 23 2012

A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 9, 12, 26, 48, 90, 172, 348, 664, 1349, 2680, 5438, 10976, 22510, 45900, 94700, 195032, 404442, 838824, 1748308, 3646368, 7632628, 15994232, 33606168, 70699504, 149050669, 314625264, 665280246, 1408436672, 2986069782, 6337988876

Offset: 0

José Luis Ramírez Ramírez, Apr 27 2015

nonn

			For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A090344, A086622, A257178, A257388, A007317, A064613, A104455, A104498, A257389.

CoefficientList[Series[(1-2*x^3-Sqrt[(1-2x^3)*(1-4*x^2-2*x^3)])/(2*x^2*(1-2*x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 28 2015 *)

a(n):=sum((binomial(2*m,m)/(m+1)*(if mod(n+m,3)=0 then 2^((n-2*m)/3)* binomial((m+n)/3,m) else 0)),m,0,n); /* Vladimir Kruchinin, Mar 07 2016 */

seq(n)={Vec((1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3) + O(x^(3+n))))/(2*x^2*(1-2*x^3)))} \\ Andrew Howroyd, May 01 2020

G.f.: (1-2*x^3-sqrt((1-2x^3)*(1-4*x^2-2*x^3)))/(2*x^2*(1-2*x^3)).

Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +(n+4)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(-6*n+17)*a(n-4) +4*(-3*n+10)*a(n-5) +4*(3*n-11)*a(n-6) +4*(11*n-50)*a(n-7) +20*(n-6)*a(n-8)=0. - R. J. Mathar, Jun 07 2016

Terms a(31) and beyond from Andrew Howroyd, May 01 2020

A349639 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^k.

Original entry on oeis.org

1, 2, 11, 163, 4177, 150606, 7002679, 399296682, 26997867705, 2112814307980, 187919721166951, 18727570061711897, 2067435790679136937, 250474099952311886236, 33043529154916822685459, 4715582224589290429430011, 723854564711343436767660481, 118933484485939500023357177356

Offset: 0

Vaclav Kotesovec, Nov 23 2021

nonn

Cf. A007317, A064613, A291699, A292632, A349603, A349640.

Table[1+Sum[Binomial[n, j]*CatalanNumber[j]*j^j, {j, 1, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k^k); \\ Michel Marcus, Nov 23 2021

a(n) ~ c * 2^(2*n) * n^(n - 3/2) /sqrt(Pi), where c = Sum_{k>=0} 1/(4^k*k!*exp(k)) = exp(exp(-1)/4) = 1.09633177846412646399584148732...

cp's OEIS Frontend

A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.

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A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

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A271025 A(n, k) is the n-th binomial transform of the Catalan sequence (A000108) evaluated at k. Array read by descending antidiagonals for n >= 0 and k >= 0.

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A171515 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033543.

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A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

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A349639 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^k.

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