A187021 -id:A187021 - cp's OEIS frontend

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

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

A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2(k+1)x + ((k-1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 20, 1, 1, 5, 22, 63, 70, 1, 1, 6, 33, 136, 321, 252, 1, 1, 7, 46, 245, 886, 1683, 924, 1, 1, 8, 61, 396, 1921, 5944, 8989, 3432, 1, 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1, 1, 10, 97, 848, 6145, 33876, 127905, 281488, 265729, 48620, 1

Offset: 0

Seiichi Manyama, May 02 2019

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) - k*x*y). - Seiichi Manyama, Jul 11 2020

More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 - (k+r-1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

			Square array begins:
  1,   1,    1,     1,      1,      1,      1, ...
  1,   2,    3,     4,      5,      6,      7, ...
  1,   6,   13,    22,     33,     46,     61, ...
  1,  20,   63,   136,    245,    396,    595, ...
  1,  70,  321,   886,   1921,   3606,   6145, ...
  1, 252, 1683,  5944,  15525,  33876,  65527, ...
  1, 924, 8989, 40636, 127905, 324556, 712909, ...
Seen as a triangle T(n, k):
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  1;
  [3] 1, 3,  6,   1;
  [4] 1, 4, 13,  20,    1;
  [5] 1, 5, 22,  63,   70,     1;
  [6] 1, 6, 33, 136,  321,   252,     1;
  [7] 1, 7, 46, 245,  886,  1683,   924,     1;
  [8] 1, 8, 61, 396, 1921,  5944,  8989,  3432,     1;
  [9] 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1;

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

Columns k=0..6 give A000012, A000984, A001850, A069835, A084771, A084772, A098659.
Main diagonal gives A187021.
T(n,n+1) gives A335309.
Cf. A307855, A307884, A307910.

# Seen as a triangle read by rows:
T := (n, k) -> simplify(hypergeom([-k, -k], [1], n - k)):
seq(lprint(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, May 13 2024

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)
(* Seen as a triangle read by rows: *)
T[n_, k_] := HypergeometricPFQ[{-k, -k}, {1}, n - k];
Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, May 13 2024 *)

T(n,k) is the coefficient of x^n in the expansion of (1 + (k+1)*x + k*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-1) * T(n-2,k).
T(n,k) = hypergeom([-k, -k], [1], n - k), (triangular form). - Detlef Meya, May 13 2024

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

A241247 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^3.

Original entry on oeis.org

2, 21, 352, 8065, 231876, 7951069, 314931968, 14095941633, 701590424500, 38358147922501, 2281458125531520, 146469277526152321, 10084388675810865248, 740560093656498673965, 57738578482070455269376, 4760258648137662340202497, 413561386818608994516491316

Offset: 1

Vaclav Kotesovec, Apr 18 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A187021, A234971.

Table[Sum[n^k*Binomial[n,k]^3,{k,0,n}],{n,1,20}]
Table[HypergeometricPFQ[{-n,-n,-n},{1,1},-n],{n,1,20}]

a(n) = sum(k=0, n, n^k*binomial(n,k)^3); \\ Michel Marcus, Jul 11 2020

a(n) ~ exp(1 - 3*n^(1/3)/2 + 3*n^(2/3)) * n^(n-2/3) / (2*Pi*sqrt(3)) * (1 + 5/(4*n^(1/3))).

A307885 Coefficient of x^n in (1 - (n-1)x - nx^2)^n.

Original entry on oeis.org

1, 0, -3, 28, -255, 2376, -20195, 71688, 3834369, -187855280, 6676401501, -220595216280, 7180102389889, -234023553073296, 7631745228481725, -245429882267144624, 7501602903392006145, -196609711096827812448, 2542435002501531333949

Offset: 0

Seiichi Manyama, May 02 2019

sign

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

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

Main diagonal of A307884.

Cf. A187021.

A307885:= n -> simplify(hypergeom([-n,-n], [1], -n));
seq(A307885(n), n = 0..30); # G. C. Greubel, May 31 2020

Table[Hypergeometric2F1[-n, -n, 1, -n], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2019 *)

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

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

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

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

a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^2.
a(n) = Sum_{k=0..n} (-n-1)^(n-k) * binomial(n,k) * binomial(n+k,k).
a(n) = Hypergeometric2F1(-n, -n, 1, -n). - Vaclav Kotesovec, May 07 2019
a(n) = n! * [x^n] exp((1 - n)*x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020

A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

Original entry on oeis.org

1, 2, 13, 352, 38401, 16971876, 29359436149, 207003074670848, 5679112509686022145, 636468045901197095750500, 277939985126193076692203962501, 494649880078824954885176565423811200, 3447375085398645453825889951638344722092289, 97424105704407389799712313421357308088296084669504

Offset: 0

Seiichi Manyama, Jul 11 2020

Seiichi Manyama, Table of n, a(n) for n = 0..57
Vaclav Kotesovec, Plot of a(n) / (2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2)) for n = 1..10000000

Main diagonal of A336187.

Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214.

[(&+[n^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Unprotect[Power]; 0^0 = 1; a[n_] := Sum[n^k * Binomial[n, k]^n, {k, 0, n} ]; Array[a, 14, 0] (* Amiram Eldar, Jul 11 2020 *)

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

[sum(n^j*binomial(n,j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Let f(n) = 2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2). For sufficiently large n 0.7675... < a(n)/f(n) < 0.7900... - Vaclav Kotesovec, Jul 11 2020
The above bounds of Vaclav Kotesovec can be recast as: |a(n)/f(n) - exp(-1/4)| <= (3*Pi)^(-2) for sufficiently large n. - Peter Luschny, Jul 12 2020
a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-2), exp(-4)) * 2^(n^2 + n/2) / Pi^(n/2) if n is even and a(n) ~ exp(-3/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-4), exp(-4)) * 2^(n^2 + n/2) * sqrt(n)  / Pi^(n/2) if n is odd. - Vaclav Kotesovec, Jul 13 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]

a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

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)

A234971 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^4.

Original entry on oeis.org

1, 2, 37, 1000, 38401, 1896876, 112124629, 7679202336, 595411451905, 51348552829300, 4861414171762501, 500163335120177136, 55466421261812540929, 6585829687114412247800, 832587068884779776276661, 111541424966889778569909376, 15771414153994526723881828353

Offset: 0

Vaclav Kotesovec, Apr 19 2014

In general, Sum_{k=0..n} n^k * binomial(n,k)^p is asymptotic to (1+n^(1/p))^(n*p+p-1) / sqrt(p * (2*Pi)^(p-1) * n^(p-1/p)).

Vincenzo Librandi, Table of n, a(n) for n = 0..200 [a(0)=1 inserted by _Georg Fischer_, Jan 04 2020]
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Cf. A187021, A241247.

a := n -> hypergeom([-n, -n, -n, -n], [1, 1, 1], n):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Dec 22 2020

Table[Sum[If[n==k==0, 1, n^k]*Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* offset adapted by Georg Fischer, Jan 04 2021 *)

a(n) = sum(k=0, n, n^k * binomial(n,k)^4); \\ Michel Marcus, Jan 04 2021

a(n) ~ (1+n^(1/4))^(4*n+3) / (4*sqrt(2) * Pi^(3/2) * n^(15/8)).

a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], n). - Peter Luschny, Dec 22 2020

a(0) = 1 prepended by Peter Luschny, Dec 22 2020

A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.

Main diagonal of A341014.

[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019

a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

cp's OEIS Frontend

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

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A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2*(k+1)*x + ((k-1)*x)^2).

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

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A241247 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^3.

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A307885 Coefficient of x^n in (1 - (n-1)*x - n*x^2)^n.

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A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

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A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2(k+1)x + ((k-1)*x)^2).

A307885 Coefficient of x^n in (1 - (n-1)x - nx^2)^n.