A247496
a(n) = n!*[x^n](exp(n*x)*BesselI_{1}(2*x)/x), n>=0, main diagonal of A247495.
Original entry on oeis.org
1, 1, 5, 36, 354, 4425, 67181, 1200745, 24699662, 574795035, 14930563042, 428235433978, 13442267711940, 458373150076335, 16872717817840509, 666835739823870900, 28163028244810505622, 1265837029802096365275, 60330098878933736719190, 3039079334694016053006276
Offset: 0
-
Flatten[{1,Table[n^n*HypergeometricPFQ[{1/2-n/2, -n/2}, {2}, 4/n^2],{n,1,20}]}] (* Vaclav Kotesovec, Dec 12 2014 *)
-
{a(n) = sum(k=0, n\2, n^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1))} \\ Seiichi Manyama, May 05 2019
-
{a(n) = polcoef((1+n*x+x^2)^(n+1)/(n+1), n)} \\ Seiichi Manyama, May 06 2019
-
a = lambda n: 1 if n==0 else n^n*hypergeometric([1/2-n/2, -n/2], [2], 4/n^2).simplify()
[a(n) for n in range(20)]
A247497
Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247495 (the Motzkin polynomials evaluated at nonnegative integers).
Original entry on oeis.org
1, 1, 1, 2, 3, 2, 4, 10, 12, 6, 9, 33, 62, 60, 24, 21, 111, 300, 450, 360, 120, 51, 378, 1412, 3000, 3720, 2520, 720, 127, 1303, 6552, 18816, 32760, 34440, 20160, 5040, 323, 4539, 30186, 113820, 264264, 388080, 352800, 181440, 40320
Offset: 0
Triangle starts:
[ 1],
[ 1, 1],
[ 2, 3, 2],
[ 4, 10, 12, 6],
[ 9, 33, 62, 60, 24],
[ 21, 111, 300, 450, 360, 120],
[ 51, 378, 1412, 3000, 3720, 2520, 720],
[127, 1303, 6552, 18816, 32760, 34440, 20160, 5040].
.
[n=3] -> [4,10,12,6] -> 4/(x-1)+10/(x-1)^2+12/(x-1)^3+6/(x-1)^4 = 2*x*(-x+2*x^2+2)/(x-1)^4; generating function of A247495[n,3] = 0,4,14, 36,...
[n=4] -> [9,33,62,60,24] -> -9/(x-1)-33/(x-1)^2-62/(x-1)^3-60/(x-1)^4-24/(x-1)^5 = -(2-x-3*x^3+17*x^2+9*x^4)/(x-1)^5; generating function of A247495[n,4] = 2,9,42,137,...
-
A247497_row := proc(n) local A, M, p;
A := (n,k) -> `if`(type(n-k, odd),0,n!/(k!*((n-k)/2)!^2*((n-k)/2+1))):
M := (k,x) -> add(A(k,j)*x^j,j=0..k): # Motzkin polynomial
p := expand(sum(x^k*M(n,k),k=0..infinity));
[seq((-1)^(n+1)*coeff(convert(p,parfrac),(x-1)^(-j)),j=1..n+1)] end:
seq(print(A247497_row(n)),n=0..7);
A306684
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, 4, 7, 10, 13, 16, 19, 22, ...
1, 9, 21, 37, 57, 81, 109, 141, ...
1, 21, 61, 121, 201, 301, 421, 561, ...
1, 51, 191, 451, 861, 1451, 2251, 3291, ...
1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...
-
T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)
A294573
a(n) = n! * [x^n] exp((n+1)*x)*BesselI(1,2*x)/x.
Original entry on oeis.org
1, 2, 10, 76, 777, 9996, 155139, 2821400, 58856963, 1385621260, 36343079188, 1051024082472, 33226817252215, 1140040324751160, 42193259673938754, 1675570154136359472, 71069261432474378715, 3206616936773061141900, 153358034674756782660342, 7749560706936442485607560
Offset: 0
-
S:= series(exp((n+1)*x)*BesselI(1,2*x)/x, x, 102):
seq(simplify(n!*coeff(S,x,n)),n=0..100); # Robert Israel, Nov 03 2017
-
Table[n! SeriesCoefficient[Exp[(n + 1) x] BesselI[1, 2 x]/x, {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[(1 - (n + 1) x - Sqrt[(1 - (n - 1) x) (1 - (n + 3) x)])/(2 x^2), {x, 0, n}], {n, 0, 19}]
Table[(n + 1)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 4/(n + 1)^2], {n, 0, 19}]
Showing 1-4 of 4 results.
Comments