A187018 -id:A187018 - 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

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

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

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

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)

A307862 Coefficient of x^n in (1 + x - n*x^2)^n.

Original entry on oeis.org

1, 1, -3, -17, 49, 651, -1259, -38023, 26433, 2969299, 2225101, -289389891, -692529551, 33718183045, 143578976997, -4559187616649, -29119975483135, 699788001188403, 6188699469443869, -119828491083854707, -1404529670244379599, 22563726025297759345, 341997845736800473397

Offset: 0

Seiichi Manyama, May 02 2019

sign

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

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

Main diagonal of A307860.

Cf. A002426, A187018, A307885.

A307862:= n -> simplify(hypergeom([-n/2, (1-n)/2], [1], -4*n));
seq(A307862(n), n = 0..30); # G. C. Greubel, May 31 2020

a[n_]:= SeriesCoefficient[(1 +x -n*x^2)^n, {x,0,n}]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 31 2020 *)

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

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

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

[ hypergeometric([-n/2, (1-n)/2], [1], -4*n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020

a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * binomial(2*k,k).
a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = Hypergeometric2F1(-n/2, (1-n)/2; 1; -4*n). - G. C. Greubel, May 31 2020

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

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

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

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A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

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

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.

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

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