A241247 -id:A241247 - 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

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

A274246 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 2^(n-k) * k!.

Original entry on oeis.org

1, 3, 22, 230, 3048, 48152, 875536, 17907024, 405320320, 10030449536, 268836428544, 7744939895552, 238352004594688, 7795463142466560, 269761049981827072, 9839883848966985728, 377091995258812268544, 15139047281589466136576, 635088889901946682408960, 27775758544209632635060224, 1263876454164193257295446016

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..417

Cf. A000172, A087912, A216831, A241247, A277386.

[(&+[Binomial(n,j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Dec 27 2022

f:= gfun:-rectoproc({n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3),a(0)=1,a(1)=3,a(2)=22},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 16 2017

Table[Sum[Binomial[n, k]^3 * 2^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

def A274246(n): return sum(binomial(n,j)^3*factorial(j)*2^(n-j) for j in range(n+1))
[A274246(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

Recurrence: n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3).
a(n) ~ n^(n - 1/6) * exp(3*2^(1/3)*n^(2/3) - 2^(2/3)*n^(1/3) - n + 2/3) / (2^(5/6)*sqrt(3*Pi)) * (1 + 31*2^(1/3)/(27*n^(1/3)) + 3437/(3645*2^(1/3) * n^(2/3))).
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 2^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022
a(n) = 2^n * Hypergeometric3F1([-n, -n, -n], [1], -1/2). - G. C. Greubel, Dec 27 2022

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

A277386 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.

Original entry on oeis.org

1, 4, 35, 438, 6873, 127488, 2703447, 64121130, 1674999009, 47638235484, 1461975938379, 48068355965886, 1683311251028265, 62477888170824792, 2447583053876363727, 100842325515959413842, 4356021203508275392833, 196739133595421931988020, 9268144156277932321747251

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

Cf. A000172, A277382, A241247, A274246.

Table[Sum[Binomial[n, k]^3 * 3^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

Recurrence: n*(8*n - 23)*a(n) = 3*(8*n^3 - 15*n^2 - 30*n + 17)*a(n-1) - (n-1)*(24*n^3 - 261*n^2 + 770*n - 666)*a(n-2) + (n-2)^3*(n-1)*(8*n - 15)*a(n-3).
a(n) ~ n^(n - 1/6) * exp(3*3^(1/3)*n^(2/3) - 3^(2/3)*n^(1/3) - n +1) / (3^(5/6)*sqrt(2*Pi)) * (1 + 19/(6*3^(2/3)*n^(1/3)) + 1193/(1080*3^(1/3) * n^(2/3))).
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 3^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

A336180 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^3.

Original entry on oeis.org

1, 0, -11, 136, -639, -25624, 1133245, -27431424, 259448833, 17402599792, -1405909697499, 63884679938960, -1830503703899519, -5324845289379264, 5494299851213052685, -496909924804074650624, 30201149245542631276545, -1236819213672144144878752, 5410434345252588202534741

Offset: 0

Seiichi Manyama, Jul 10 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..375
Vaclav Kotesovec, Plot of |a(n)/a(n-1)|/n for n = 1..1000

Main diagonal of A336179.

Cf. A241247, A307885.

a := n -> hypergeom([-n, -n, -n], [1, 1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020

Array[Function[n, 1 + Sum[(-n)^k Binomial[n, k]^3, {k, n}]], 19, 0] (* Jan Mangaldan, Jul 14 2020 *)

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

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

A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 10, 1, 1, 4, 21, 56, 1, 1, 5, 34, 171, 346, 1, 1, 6, 49, 352, 1521, 2252, 1, 1, 7, 66, 605, 3946, 14283, 15184, 1, 1, 8, 85, 936, 8065, 46744, 138909, 104960, 1, 1, 9, 106, 1351, 14346, 113525, 573616, 1385163, 739162, 1, 1, 10, 129, 1856, 23281, 231876, 1656145, 7217536, 14072193, 5280932, 1

Offset: 0

Seiichi Manyama, Jul 10 2020

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z).

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z).

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,   10,    21,    34,     49,     66, ...
  1,   56,   171,   352,    605,    936, ...
  1,  346,  1521,  3946,   8065,  14346, ...
  1, 2252, 14283, 46744, 113525, 231876, ...

Columns k=0-6 give: A000012, A000172, A206178, A206180, A216483, A216636, A216698.
Main diagonal gives A241247.
Cf. A307883, A336179, A336187.

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[k^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

Original entry on oeis.org

1, 2, 22, 438, 12824, 496370, 23914512, 1379269094, 92667551104, 7108231236066, 612974464428800, 58702772664490262, 6181602019316333568, 709911177607125141362, 88301595129435811723264, 11825985945777638231211750, 1696696168760520436580974592, 259624546758869333450285984066

Offset: 0

Ilya Gutkovskiy, Jun 12 2022

nonn

Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k)^3 * k! * n^(n-k)); \\ Michel Marcus, Jun 12 2022

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.

a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - Vaclav Kotesovec, Jul 01 2022, updated Mar 17 2024

cp's OEIS Frontend

A187021 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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A274246 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 2^(n-k) * k!.

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

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A277386 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.

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A336180 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^3.

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A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

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