A092366 -id:A092366 - cp's OEIS frontend

A360592 G.f.: Sum_{k>=0} (1 + kx)^k x^k.

1, 1, 2, 5, 14, 44, 149, 543, 2096, 8539, 36444, 162380, 752181, 3612037, 17933038, 91843329, 484280386, 2624400428, 14595111277, 83178971707, 485218783724, 2893881790823, 17628815344600, 109585578277012, 694575012732989, 4485139961090153, 29486515600393930

Offset: 0

Vaclav Kotesovec, Feb 13 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..760

Cf. A360232, A092366, A187018, A360699, A360707.

N:= 40:
S:= series(add((1+k*x)^k*x^k, k=0..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Feb 13 2023

nmax = 30; CoefficientList[Series[Sum[(1 + k*x)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x]
Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^k, {k, 0, n/2}], {n, 1, 30}]}]

{a(n) = polcoeff(sum(m=0, n, (1 + m*x)^m * x^m + x*O(x^n)), n)};
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * (n-k)^k.

a(n) ~ exp(exp(1/2)*sqrt(n/2) - 3*exp(1)/8) * n^(n/2) / 2^(n/2 + 1) * (1 + ((exp(1/2) + exp(-1/2))/2^(5/2) + 11*exp(3/2)/2^(9/2))/sqrt(n)).

A186925 Coefficient of x^n in (1+n*x+x^2)^n.

Original entry on oeis.org

1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795, 83829045032101462204100, 4229207755493569286374167

Offset: 0

Emanuele Munarini, Mar 02 2011

Seiichi Manyama, Table of n, a(n) for n = 0..386 (terms 0..100 from Vincenzo Librandi)

Main diagonal of A292627.

Cf. A092366, A187018, A187019, A187021, A070910, A292629.

P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011

Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
Table[SeriesCoefficient[(1 + n*x + x^2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2023 *)

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

makelist(ultraspherical(n,-n,-n/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */
makelist(a(n),n,0,20);

{a(n) = sum(k=0, n, (n-2)^(n-k)*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, May 01 2019

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

a(n) = [x^n] (1+n*x+x^2)^n.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).
a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014
a(n) = GegenbauerPoly(n,-n,-n/2). - Emanuele Munarini, Oct 20 2016
From Ilya Gutkovskiy, Sep 20 2017: (Start)
a(n) = [x^n] 1/sqrt((1 + 2*x - n*x)*(1 - 2*x - n*x)).
a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)
From Seiichi Manyama, May 01 2019: (Start)
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (n+2)^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). (End)
a(n) = (1/4)^n * Sum_{k=0..n} (n-2)^k * (n+2)^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

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

Original entry on oeis.org

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

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

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

A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

Original entry on oeis.org

1, 2, 11, 100, 1257, 20076, 387739, 8766248, 226739489, 6595646860, 212944033051, 7550600079672, 291527929539433, 12169325847587832, 545918747361417291, 26183626498897556176, 1336713063706757646465

Offset: 1

Ralf Stephan, Oct 09 2004

A diagonal of Narayana array (A008550).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012

Cf. A092366, A187018, A187019, A187021.

Cf. A008550, A204057, A243631.

A099169:= func< n | (&+[Binomial(n, j)*Binomial(n-1,j)*(n-1)^j/(j+1): j in [0..n-1]]) >;
[A099169(n): n in [1..30]]; // G. C. Greubel, Feb 16 2021

A099169:= n-> add( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1), j=0..n-1);
seq( A099169(n), n=1..30) # G. C. Greubel, Feb 16 2021

Join[{1},Table[Sum[Binomial[n,k]Binomial[n,k+1](n-1)^k,{k,0,n-1}]/n,{n,2,20}]] (* Harvey P. Dale, Oct 07 2013 *)
Table[Hypergeometric2F1[1-n,-n,2,-1+n],{n,1,20}] (* Vaclav Kotesovec, Apr 18 2014 *)

a(n) = (1/n) * sum(k=0, n-1, binomial(n,k) * binomial(n,k+1) * (n-1)^k); \\ Michel Marcus, Feb 16 2021

def A099169(n): return sum( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1) for j in [0..n-1])
[A099169(n) for n in [1..30]] # G. C. Greubel, Feb 16 2021

From Vaclav Kotesovec, Apr 18 2014, extended Dec 01 2021: (Start)
a(n) = Hypergeometric2F1([1-n,-n], [2], -1+n).
a(n) ~ exp(2*sqrt(n)-2) * n^(n-7/4) / (2*sqrt(Pi)) * (1 + 119/(48*sqrt(n))). (End)

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

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

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

A307911 Coefficient of x^n in expansion of (1 - nx - nx^2)^n.

Original entry on oeis.org

1, -1, 0, 27, -416, 5625, -74304, 924385, -8626176, -48361131, 7124800000, -340421390199, 13686496542720, -522760216822129, 19658830846298112, -735037915447265625, 27218267709730979840, -980444996625142158435, 32830565919734078521344, -889052809376495994642527

Offset: 0

Seiichi Manyama, May 05 2019

sign

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

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

Main diagonal of A307819.

Cf. A092366, A307862.

a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * Binomial[2*k, k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)

PARI
```
{a(n) = polcoef((1-n*x-n*x^2)^n, n)}
```

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

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

a(n) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,k) * binomial(n-k,k) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,2*k) * binomial(2*k,k).

For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 1, -4/n). - Vaclav Kotesovec, May 12 2021

A307844 Constant term in the expansion of (n/x + 1 + n*x)^n.

Original entry on oeis.org

1, 1, 9, 55, 1729, 19251, 1050841, 16977129, 1322929665, 28017221059, 2839212609001, 74390784295653, 9283240524317761, 289865990675075725, 42976734096778661817, 1557837326400792009751, 267561369300137776050177, 11042876765198762014337235

Offset: 0

Seiichi Manyama, May 01 2019

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..351

Main diagonal of A307847.

Cf. A002426, A092366, A186925, A187018.

Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n,k] * Binomial[2*k,k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 02 2019 *)

PARI
```
{a(n) = polcoef((n/x+1+n*x)^n, 0)}
```
PARI
```
{a(n) = polcoef((1+x+(n*x)^2)^n, n)}
```

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

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

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

a(n) = Sum_{k=0..n} (1-2*n)^(n-k) * n^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (1+2*n)^(n-k) * (-n)^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..floor(n/2)} n^(2*k) *binomial(n,2*k) * binomial(2*k,k).
a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, May 02 2019

A307903 Coefficient of x^n in (1 + nx + nx^3)^n.

Original entry on oeis.org

1, 1, 4, 36, 448, 6875, 124956, 2624293, 62537728, 1667191653, 49158400000, 1588285928306, 55796298391296, 2117279603738494, 86299754734693696, 3760031421065559375, 174374733095888748544, 8575617145497637681301, 445758339115421869936896, 24417549315693295193935516

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Cf. A092366, A116411, A307904, A307905.

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

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

{a(n) = sum(k=0, n\3, n^(n-2*k)*binomial(n,3*k)*binomial(3*k,k))}

a(n) = Sum_{k=0..floor(n/3)} n^(n-2*k) * binomial(n,3*k) * binomial(3*k,k).

a(n) ~ exp(3*n^(1/3)/2^(2/3)) * n^(n - 1/6) / (2^(2/3)*sqrt(3*Pi)) * (1 - 79/(36 * 2^(1/3) * n^(1/3))). - Vaclav Kotesovec, May 05 2019

cp's OEIS Frontend

A360592 G.f.: Sum_{k>=0} (1 + k*x)^k * x^k.

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A186925 Coefficient of x^n in (1+n*x+x^2)^n.

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A187021 Coefficient of x^n in (1 + (n+1)*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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A187019 Coefficient of x^n in expansion of (1+n*x+(n+1)*x^2)^n.

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A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

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PARI

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