A247496 -id:A247496 - cp's OEIS frontend

A292716 a(n) = [x^n] 1/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - ...))))), a continued fraction.

1, 1, 6, 54, 672, 10625, 203256, 4554697, 116842496, 3373056027, 108134200000, 3809118341028, 146170521796608, 6066719073261639, 270692733123460480, 12917478278285156250, 656311833287586742272, 35364920064570086779227, 2014028255250518880457728, 120852950097737555898105210

Offset: 0

Ilya Gutkovskiy, Sep 21 2017

nonn

Also coefficient of x^n in the expansion of 1/(n+1) * (1 + n*x + n*x^2)^(n+1). - Seiichi Manyama, May 06 2019

Alois P. Heinz, Table of n, a(n) for n = 0..381

Main diagonal of A107267.

Cf. A001006, A247496.

Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-n x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[2/(1 - n x + Sqrt[1 + n x ((n - 4) x - 2)]), {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[E^(n x) Hypergeometric0F1Regularized[2, n x^2], {x, 0, n}], {n, 0, 19}]
Flatten[{1, Table[Sum[Binomial[k+1, n-k+1] * Binomial[n, k] * n^k / (k+1), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 05 2019 *)

{a(n) = polcoef((1+n*x+n*x^2)^(n+1)/(n+1), n)} \\ Seiichi Manyama, May 06 2019

a(n) = [x^n] 2/(1 - n*x + sqrt(1 + n*x*((n - 4)*x - 2))).
a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(n)*x)/(sqrt(n)*x), for n > 0.
a(n) = A107267(n,n).
a(n) ~ exp(2*sqrt(n) - 2) * n^(n - 3/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, May 05 2019

A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k](exp(nx)* BesselI_{1}(2*x)/x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 2, 4, 5, 3, 1, 0, 9, 14, 10, 4, 1, 5, 21, 42, 36, 17, 5, 1, 0, 51, 132, 137, 76, 26, 6, 1, 14, 127, 429, 543, 354, 140, 37, 7, 1, 0, 323, 1430, 2219, 1704, 777, 234, 50, 8, 1, 42, 835, 4862, 9285, 8421, 4425, 1514, 364, 65, 9, 1

Offset: 0

Peter Luschny, Dec 11 2014

This two-dimensional array of numbers can be seen as a generalization of the Motzkin numbers A001006 for two reasons: The case n=1 reduces to the Motzkin numbers and the columns are the values of the Motzkin polynomials M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j evaluated at the nonnegative integers.

			Square array starts:
[n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]      [8]
[0]   1, 0,  1,   0,    2,    0,     5,      0,      14, ...  A126120
[1]   1, 1,  2,   4,    9,   21,    51,    127,     323, ...  A001006
[2]   1, 2,  5,  14,   42,  132,   429,   1430,    4862, ...  A000108
[3]   1, 3, 10,  36,  137,  543,  2219,   9285,   39587, ...  A002212
[4]   1, 4, 17,  76,  354, 1704,  8421,  42508,  218318, ...  A005572
[5]   1, 5, 26, 140,  777, 4425, 25755, 152675,  919139, ...  A182401
[6]   1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, ...  A025230
A000012,A001477,A002522,A079908, ...
.
Triangular array starts:
              1,
             0, 1,
           1, 1, 1,
          0, 2, 2, 1,
        2, 4, 5, 3, 1,
      0, 9, 14, 10, 4, 1,
   5, 21, 42, 36, 17, 5, 1,
0, 51, 132, 137, 76, 26, 6, 1.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Main diagonal gives A247496.

Cf. A126120, A001006, A000108, A002212, A005572, A182401, A025230, A002522, A079908, A055151, A097610, A247497, A306684.

# RECURRENCE
T := proc(n,k) option remember; if k=0 then 1 elif k=1 then n else
(n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) fi end:
seq(print(seq(T(n,k),k=0..9)),n=0..6);
# OGF (row)
ogf := n -> (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2):
seq(print(seq(coeff(series(ogf(n),x,12),x,k),k=0..9)),n=0..6);
# EGF (row)
egf := n -> exp(n*x)*hypergeom([],[2],x^2):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..9)),n=0..6);
# MOTZKIN polynomial (column)
A097610 := proc(n,k) if type(n-k,odd) then 0 else n!/(k!*((n-k)/2)!^2* ((n-k)/2+1)) fi end: M := (k,x) -> add(A097610(k,j)*x^j,j=0..k):
seq(print(seq(M(k,n),n=0..9)),k=0..6);
# OGF (column)
col := proc(n, len) local G; G := A247497_row(n); (-1)^(n+1)* add(G[k+1]/(x-1)^(k+1), k=0..n); seq(coeff(series(%, x, len+1),x,j), j=0..len) end: seq(print(col(n,8)), n=0..6); # Peter Luschny, Dec 14 2014

T[0, k_] := If[EvenQ[k], CatalanNumber[k/2], 0];
T[n_, k_] := n^k*Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4/n^2];
Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

def A247495(n,k):
    if n==0: return(k//2+1)*factorial(k)/factorial(k//2+1)^2 if is_even(k) else 0
    return n^k*hypergeometric([(1-k)/2,-k/2],[2],4/n^2).simplify()
for n in (0..7): print([A247495(n,k) for k in range(11)])

T(n,k) = (n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) for k>=2.
T(n,k) = Sum_{j=0..floor(k/2)} n^(k-2*j)*binomial(k,2*j)*binomial(2*j,j)/(j+1).
T(n,k) = n^k*hypergeom([(1-k)/2,-k/2], [2], 4/n^2) for n>0.
T(n,n) = A247496(n).
O.g.f. for row n: (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2).
O.g.f. for row n: R(x)/x where R(x) is series reversion of x/(1+n*x+x^2).
E.g.f. for row n: exp(n*x)*hypergeom([],[2],x^2).
O.g.f. for column k: the k-th column consists of the values of the k-th Motzkin polynomial M_{k}(x) evaluated at x = 0,1,2,...; M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j = sum_{j=0..k} (-1)^j*binomial(k,j)*A001006(j)*(x+1)^(k-j).
O.g.f. for column k: sum_{j=0..k} (-1)^(k+1)*A247497(k,j)/(x-1)^(j+1). - Peter Luschny, Dec 14 2014
O.g.f. for row n: 1/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
T(n,k) is the coefficient of x^k in the expansion of 1/(k+1) * (1 + n*x + x^2)^(k+1). - Seiichi Manyama, May 07 2019

A307906 Coefficient of x^n in 1/(n+1) * (1 + x + n*x^2)^(n+1).

Original entry on oeis.org

1, 1, 3, 10, 57, 301, 2251, 15583, 138209, 1153603, 11592451, 111381348, 1235739385, 13276480803, 159935056555, 1884023828326, 24356065951617, 310189106485419, 4266048524240323, 58124516559463590, 844705360693479801, 12213285476055278959, 186543178982826381387

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Also coefficient of x^n in the expansion of 2/(1 - x + sqrt(1 - 2*x + (1 - 4*n)*x^2)).

Seiichi Manyama, Table of n, a(n) for n = 0..500

Main diagonal of A306684.

Cf. A000108, A001006, A187018, A247496, A292716.

Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, 4*n], {n, 0, 20}] (* Vaclav Kotesovec, May 05 2019 *)

{a(n) = polcoef((1+x+n*x^2)^(n+1)/(n+1), n)}

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

{a(n) = sum(k=0, n\2, n^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}

a(n) = [x^n]  (1 - x - sqrt(1 - 2*x + (1 - 4*n)*x^2))/(2*n*x^2).
a(n) = Sum_{k=0..floor(n/2)} n^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} n^k * binomial(n,2*k) * A000108(k).
a(n) ~ exp(sqrt(n)/2 - 1/8) * 2^(n + 1/2) * n^((n-3)/2) / sqrt(Pi). - Vaclav Kotesovec, May 05 2019

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

Ilya Gutkovskiy, Nov 02 2017

nonn

The n-th term of the n-th binomial transform of A001006.

Robert Israel, Table of n, a(n) for n = 0..300
N. J. A. Sloane, Transforms

Diagonal of A247495.

Cf. A001006, A247496.

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}]

a(n) = [x^n] (1 - (n + 1)*x - sqrt((1 - (n - 1)*x)*(1 - (n + 3)*x)))/(2*x^2).

a(n) ~ exp(1) * BesselI(1,2) * n^n. - Vaclav Kotesovec, Nov 13 2017

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).

Original entry on oeis.org

1, 1, 6, 45, 456, 5825, 89896, 1627437, 33822944, 793783233, 20765009344, 599157626925, 18904594000128, 647524807918209, 23929038677825152, 948995910652193325, 40203601321988822528, 1812025020244371552897, 86577002960871477916672, 4371100278517527047687213

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A001003, A025235, A068764, A071356, A151374, A247496, A292716.

Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]

a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).
a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).
a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).
a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017

A302286 a(n) = [x^n] 1/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 12, 116, 1530, 25422, 507696, 11814728, 313426350, 9324499610, 307171539576, 11091813369276, 435408606414964, 18453269887229478, 839464708754178240, 40786587211854543120, 2107367668847505288726, 115352793604678609311282, 6667002839420189781109800, 405656528458830256952396420

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

Main diagonal of A247507.

Cf. A006318, A099169, A247496, A292798.

Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-x, 1 - n x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - (2 n + 4) x + n^2 x^2])/(2 x), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[(1/n) Sum[(n + 1)^k Binomial[n, k] Binomial[n, k - 1], {k, 0, n}], {n, 1, 19}]]
Table[(n + 1) Hypergeometric2F1[1 - n, -n, 2, n + 1], {n, 0, 19}]

a(n) = [x^n] (1 - n*x - sqrt(1 - (2*n + 4)*x + n^2*x^2))/(2*x).
a(0) = 1; a(n) = (1/n)*Sum_{k=0..n} (n + 1)^k*binomial(n,k)*binomial(n,k-1).
a(n) = A247507(n,n).
a(n) ~ exp(2*sqrt(n)) * n^(n - 3/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Jun 08 2019

cp's OEIS Frontend

A292716 a(n) = [x^n] 1/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - ...))))), a continued fraction.

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A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k](exp(n*x)* BesselI_{1}(2*x)/x), n>=0, k>=0.

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A307906 Coefficient of x^n in 1/(n+1) * (1 + x + n*x^2)^(n+1).

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A294573 a(n) = n! * [x^n] exp((n+1)*x)*BesselI(1,2*x)/x.

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A294642 a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(2)*x)/(sqrt(2)*x).

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A302286 a(n) = [x^n] 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction.

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A292716 a(n) = [x^n] 1/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - ...))))), a continued fraction.

A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k](exp(nx)* BesselI_{1}(2*x)/x), n>=0, k>=0.

A294573 a(n) = n! * [x^n] exp((n+1)x)BesselI(1,2*x)/x.

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).

A302286 a(n) = [x^n] 1/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - ...))))), a continued fraction.