A186925 -id:A186925 - cp's OEIS frontend

A187021 Coefficient of x^n in (1 + (n+1)x + nx^2)^n.

1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101

Offset: 0

Emanuele Munarini, Mar 02 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381 (terms 0..100 from Vincenzo Librandi)
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012

Main diagonal of A307883.

Cf. A092366, A186925, A187018, A187019, A241247, A234971.

P:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011

A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) );
1, seq(A187021(n), n = 1..30); # G. C. Greubel, May 31 2020
a := n -> hypergeom([-n, -n], [1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020

Flatten[{1,Table[Sum[Binomial[n,k]^2*n^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n==0, 1, Simplify[n^(n/2)*GegenbauerC[n, -n, -(n+1)/(2 Sqrt[n])]]], {n, 0, 30}] (* Emanuele Munarini, Oct 20 2016 *)

a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n),x,n);
makelist(a(n),n,0,20);

{a(n)=sum(k=0,n,binomial(n,k)^2*n^k)} \\ Paul D. Hanna, Mar 29 2011

[1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

a(n) = [x^n] (1 + (n+1)*x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - Paul D. Hanna, Mar 29 2011
a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-1) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014
a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = hypergeom([-n, -n], [1], n). - Peter Luschny, Dec 22 2020

A092366 Coefficient of x^n in expansion of (1+nx+nx^2)^n.

Original entry on oeis.org

1, 1, 8, 81, 1120, 19375, 400896, 9630411, 262955008, 8032730715, 271175200000, 10017828457483, 401738097475584, 17371952344599385, 805429080795852800, 39844314853048828125, 2094272851244149112832, 116526044312704751752451

Offset: 0

Jon Perry, Mar 19 2004

nonn

Also coefficient of x^n in expansion of (1-2*n*x+(n^2-4*n)*x^2)^(-1/2). - Vladeta Jovovic, Mar 22 2004

Seiichi Manyama, Table of n, a(n) for n = 0..381 (terms 1..100 from Vincenzo Librandi)

Cf. A186925, A187018.

P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+n*x^2)^n)[n+1]: n in [1..22] ]; // Klaus Brockhaus, Mar 03 2011

seq(n!*coeff(series(exp(n*x)*BesselI(0,2*sqrt(n)*x),x,n+1),x,n),n=1..17);

Table[Sum[n^k*Binomial[n,k]*Binomial[k,n-k],{k,Floor[n/2],n}],{n,1,20}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n == 0, 1, n^(n/2) GegenbauerC[n, -n, -Sqrt[n]/2]], {n, 0,
12}] (* Emanuele Munarini, Oct 20 2016 *)

a(n):=coeff(expand((1+n*x+n*x^2)^n), x, n);
makelist(a(n), n, 1, 12); /* Emanuele Munarini, Mar 02 2011 */

q(n)=(1+n*x+n*x^2)^n; for(i=0,20,print1(","polcoeff(q(i),i)))

a(n) = n^(n/2)*GegenbauerPoly(n,-n,-sqrt(n)/2). - Emanuele Munarini, Oct 20 2016
Sum_{k=floor(n/2)..n} n^k*binomial(n, k)*binomial(k, n-k). - Vladeta Jovovic, Mar 22 2004
a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014

a(0)=1 prepended by Seiichi Manyama, May 01 2019

A187018 Coefficient of x^n in (1 + x + n*x^2)^n.

Original entry on oeis.org

1, 1, 5, 19, 145, 851, 7741, 58605, 600769, 5420035, 61026901, 628076153, 7648488145, 87388647373, 1138801242125, 14182492489651, 196218339243777, 2628971539313875, 38377805385510181, 547815690902283225, 8395817775835635601, 126725586542235932329

Offset: 0

Emanuele Munarini, Mar 02 2011

			G.f. = 1 + x + 5*x^2 + 19*x^3 + 145*x^4 + 851*x^5 + 7741*x^6 + 58605*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..590 (terms 0..122 from Vincenzo Librandi)

Cf. A092366, A186925, A187019, A187021.

P:=PolynomialRing(Integers()); [ Coefficients((1+x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011

A187018:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -1/(2*sqrt(n))) );
1, seq(A187018(n), n = 1..30); # G. C. Greubel, May 31 2020

Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^k, {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
a[ n_]:= SeriesCoefficient[ (1 + x + n*x^2)^n, {x, 0, n}]; (* Michael Somos, Dec 12 2014 *)
Table[If[n == 0, 1, Simplify[n^(n/2) GegenbauerC[n, -n, -1/(2 Sqrt[n])]]], {n, 0, 12}] (* Emanuele Munarini, Oct 20 2016 *)

a(n):=coeff(expand((1+x+n*x^2)^n),x,n);
makelist(a(n),n,0,20);

{a(n)=polcoeff(1/sqrt(1 - 2*x - (4*n-1)*x^2 +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Dec 12 2014

{a(n) = polcoef((1+x+n*x^2)^n, n)} \\ Seiichi Manyama, May 01 2019

[1]+[ n^(n/2)*gegenbauer(n, -n, -1/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

a(n) = [x^n] (1 + x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-1/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^k.
a(n) ~ 2^(n-1/2) * exp(sqrt(n)/2-1/8) * n^(n/2-1/2) / sqrt(Pi). - Vaclav Kotesovec, Apr 17 2014
a(n) = [x^n] 1/sqrt(1 - 2*x - (4*n-1)*x^2). - Paul D. Hanna, Dec 12 2014
a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020

A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(0,2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 3, 6, 7, 6, 1, 4, 11, 20, 19, 0, 1, 5, 18, 45, 70, 51, 20, 1, 6, 27, 88, 195, 252, 141, 0, 1, 7, 38, 155, 454, 873, 924, 393, 70, 1, 8, 51, 252, 931, 2424, 3989, 3432, 1107, 0, 1, 9, 66, 385, 1734, 5775, 13236, 18483, 12870, 3139, 252, 1, 10, 83, 560, 2995, 12276, 36645, 73392, 86515, 48620, 8953, 0

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A126869 evaluated at n.

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + (k^2 + 2)*x^2/2! + (k^3 + 6*k)*x^3/3! + (k^4 + 12*k^2 + 6)*x^4/4! + (k^5 + 20*k^3 + 30*k)*x^5/5! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  0,   1,    2,    3,     4,     5,  ...
  2,   3,    6,   11,    18,    27,  ...
  0,   7,   20,   45,    88,   155,  ...
  6,  19,   70,  195,   454,   931,  ...
  0,  51,  252,  873,  2424,  5775,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0..7 give A126869, A002426, A000984, A026375, A081671, A098409, A098410, A104454.
Rows n=0..2 give A000012, A001477, A059100.
Main diagonal gives A186925.
Cf. A292628, A307847.

Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/Sqrt[(1 + 2 x - k x) (1 - 2 x - k x)], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
From Seiichi Manyama, May 01 2019: (Start)
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(n,2*j) * binomial(2*j,j). - Seiichi Manyama, May 04 2019
T(n,k) =  (1/Pi) * Integral_{x = -1..1} (k - 2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (k + 2 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
A(n,k) = (1/4)^n * Sum_{j=0..n} (k-2)^j * (k+2)^(n-j) * binomial(2*j,j) * binomial(2*(n-j),n-j). - Seiichi Manyama, Aug 18 2025

A247496 a(n) = n![x^n](exp(nx)BesselI_{1}(2x)/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

Peter Luschny, Dec 12 2014

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A001006, A186925, A247495, A292716.

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)]

a(n) = Sum_{j=0..floor(n/2)} ((j+1)*n^(n-2*j)*n!)/((j+1)!^2*(n-2*j)!).
a(n) ~ BesselI(1,2) * n^n. - Vaclav Kotesovec, Dec 12 2014
From Ilya Gutkovskiy, Sep 21 2017: (Start)
a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 4)*x^2))/(2*x^2).
a(n) = [x^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. (End)

A187019 Coefficient of x^n in expansion of (1+nx+(n+1)x^2)^n.

Original entry on oeis.org

1, 1, 10, 99, 1366, 23525, 484436, 11582375, 314830342, 9576682569, 322014499852, 11851803991115, 473634489404220, 20414267521982893, 943592267071798696, 46545155813085562575, 2439857423310573714758

Offset: 0

Emanuele Munarini, Mar 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A092366, A186925, A187018.

P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+(n+1)*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011

Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k)*(n+1)^k, {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 18 2014 *)
Flatten[{1,Table[n^n * Hypergeometric2F1[1/2-n/2,-n/2,1,4*(1+n)/n^2],{n,1,20}]}] (* Vaclav Kotesovec, Apr 18 2014 *)

a(n):=coeff(expand((1+n*x+(n+1)*x^2)^n),x,n);
makelist(a(n),n,0,12);

a(n) = polcoef((1+n*x+(n+1)*x^2)^n, n); \\ Michel Marcus, Jun 01 2020

a(n) = [x^n] (1+n*x+(n+1)*x^2)^n.
a(n) = Sum (C(n, k)*C(n-k, n-2*k)*n^(n-2*k)*(n+1)^k, k=0..floor(n/2)).
a(n) ~ exp(2*sqrt(n)-2) * n^(n-1/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 18 2014
a(n) = n! * [x^n] exp(n*x) * BesselI(0,2*sqrt(n + 1)*x). - Ilya Gutkovskiy, Jun 01 2020

A292629 a(n) = n! * [x^n] exp(nx)BesselI(1,2*x).

Original entry on oeis.org

0, 1, 4, 30, 304, 3885, 59976, 1085973, 22571200, 529712667, 13856212600, 399773871068, 12612288989664, 431948624278795, 15960564546516240, 632898895109081310, 26809122466181751552, 1208177444352064438155, 57719104861915100554200, 2913802658820378870546498, 154991214138728849712151200

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

nonn

The n-th term of the n-th binomial transform of A138364.

Seiichi Manyama, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Main diagonal of A292628.

Cf. A138364, A186925.

Table[n!*SeriesCoefficient[E^(n*x)*BesselI[1,2*x],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)

a(n) = polcoef((1+n*x+x^2)^n, n-1); \\ Michel Marcus, May 01 2019

a(n) = A292628(n,n).
a(n) ~ BesselI(1,2) * n^n. - Vaclav Kotesovec, Sep 20 2017
a(n) = [x^(n-1)] (1+n*x+x^2)^n = [x^(n+1)] (1+n*x+x^2)^n. - Seiichi Manyama, May 01 2019

A188202 Central coefficients in (1 + 2^n*x + x^2)^n.

Original entry on oeis.org

1, 2, 18, 560, 68614, 34210752, 69223161876, 564393502852608, 18462508476328312902, 2418515748615522678533120, 1267759405004680431879193624828, 2658525712652053771685500828954042368

Offset: 0

Paul D. Hanna, Mar 24 2011

nonn

Variant of the central trinomial coefficients: A002426(n) = [x^n] (1+x+x^2)^n.

			Ignoring the initial term, this sequence forms the logarithmic series:
L(x) = 2*x + 18*x^2/2 + 560*x^3/3 + 68614*x^4/4 + 34210752*x^5/5 + ...
where the g.f. of A188203 begins:
exp(L(x)) = 1 + 2*x + 11*x^2 + 206*x^3 + 17586*x^4 + 6878604*x^5 + ...
Illustrate definition.
The coefficients of x^k in (1 + 2^n*x + x^2)^n, k=0..2n, n>=0, begin:
n=0: [(1)];
n=1: [1, (2), 1];
n=2: [1, 8, (18), 8, 1];
n=3: [1, 24, 195, (560), 195, 24, 1];
n=4: [1, 64, 1540, 16576, (68614), 16576, 1540, 64, 1];
n=5: [1, 160, 10245, 328320, 5273610, (34210752), 5273610, 328320, ...];
n=6: [1, 384, 61446, 5244800, 251904015, 6458183424, (69223161876), ...];
n=7: [1, 896, 344071, 73405696, 9396961301, 721848120448, 30814514741283, (564393502852608), ...]; ...
where the above central coefficients in parenthesis form this sequence.

G. C. Greubel, Table of n, a(n) for n = 0..57

Cf. A188203 (exp); variants: A002426, A186925.

Table[Sum[Binomial[n,k] * Binomial[n-k, n-2k] * 2^(n*(n-2k)), {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Feb 11 2015 *)

PARI
```
{a(n)=polcoeff((1+2^n*x+x^2)^n,n)}
```

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

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(n-k, n-2k) * 2^(n*(n-2k)).
Equals the logarithmic derivative of A188203 (ignoring initial term).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Feb 12 2015

A307905 Coefficient of x^n in (1 + n*x + x^3)^n.

Original entry on oeis.org

1, 1, 4, 30, 304, 3875, 59631, 1076383, 22309120, 522262245, 13631508400, 392535959156, 12362973152751, 422774554883590, 15600699362473876, 617888566413340503, 26145122799198386944, 1177107512023013681429, 56185125998674634494980, 2834081165961033246374350

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..385

Cf. A116411, A186925, A307903, A307904.

f:= n -> coeff((1+n*x+x^3)^n,x,n):
map(f, [$0..30]); # Robert Israel, Mar 27 2023

Flatten[{1, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27/(4*n^3)], {n, 1, 20}]}] (* Vaclav Kotesovec, May 05 2019 *)

PARI
```
{a(n) = polcoef((1+n*x+x^3)^n, n)}
```

{a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n,3*k)*binomial(3*k,k))}

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

a(n) ~ c * n^n, where c = Sum_{k>=0} 1/(k!*(2*k)!) = HypergeometricPFQ[{}, {1/2, 1}, 1/4] = 1.52106585051363080966025715155941607334728986626976774617... - Vaclav Kotesovec, May 05 2019

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^kx^k/k] log( Sum_{m>=0} (1 + my + y^2)^m * x^m ) for k >= 1.

Original entry on oeis.org

1, 1, 5, 38, 391, 5077, 79535, 1458264, 30621237, 724555611, 19076629520, 553236991215, 17525729241605, 602215048797900, 22312035980459259, 886733059906749795, 37631474149766344476, 1698581174869953607957, 81257725943229600518977, 4106922637708383448243974

Offset: 0

Paul D. Hanna, Feb 12 2023

nonn

Related series: M(x) = exp( Sum_{k>=1} A002426(k) * x^k/k ), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006) and A002426(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for k >= 1.

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + 724555611*x^9 + ...
such that
log(A(x)) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + ... + A360348(n)*x^n/n + ...
where A360348(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A360348, A360239, A186925, A001006, A002426.

{A360348(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1,n, A360348(m)*x^m/m ) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ BesselI(0, 2) * n^n. - Vaclav Kotesovec, Feb 12 2023

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A187021 Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.

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A092366 Coefficient of x^n in expansion of (1+n*x+n*x^2)^n.

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A187018 Coefficient of x^n in (1 + x + n*x^2)^n.

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A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(0,2*x).

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A247496 a(n) = n!*[x^n](exp(n*x)*BesselI_{1}(2*x)/x), n>=0, main diagonal of A247495.

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A187019 Coefficient of x^n in expansion of (1+n*x+(n+1)*x^2)^n.

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A187021 Coefficient of x^n in (1 + (n+1)x + nx^2)^n.

A092366 Coefficient of x^n in expansion of (1+nx+nx^2)^n.

A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(0,2*x).

A247496 a(n) = n![x^n](exp(nx)BesselI_{1}(2x)/x), n>=0, main diagonal of A247495.

A187019 Coefficient of x^n in expansion of (1+nx+(n+1)x^2)^n.

A292629 a(n) = n! * [x^n] exp(nx)BesselI(1,2*x).

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^kx^k/k] log( Sum_{m>=0} (1 + my + y^2)^m * x^m ) for k >= 1.