A005725 -id:A005725 - cp's OEIS frontend

A104312 Prime coefficient of x^n in (x^3+x^2+x+1)^n for n in A104311.

3, 31, 101, 3823, 2266366724843687556556015073508073201681

Offset: 1

T. D. Noe, Mar 01 2005

nonn

a(6), which corresponds to n=649, is too large to include.

Cf. A005725 (quadrinomial coefficients), A104314 (nontrivial prime pentanomial coefficients).

f=1; Do[f=Expand[f*(x^3+x^2+x+1)]; s=Coefficient[f, x, n]; If[PrimeQ[s], Print[{n, s}]], {n, 1000}]

A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252, 462, 1, 7, 28, 84, 210, 462, 924, 1716, 1, 8, 36, 120, 330, 792, 1716, 3432, 6427, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12861, 24229, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24300, 48520, 91828

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 20 2012

For k <= 6, the triangle coincides with triangle A213745.

			Triangle begins
n/k |  0     1     2     3     4     5     6     7     8
----+---------------------------------------------------
  0 |  1
  1 |  1     1
  2 |  1     2     3
  3 |  1     3     6    10
  4 |  1     4    10    20    35
  5 |  1     5    15    35    70   126
  6 |  1     6    21    56   126   252   462
  7 |  1     7    28    84   210   462   924  1716
  8 |  1     8    36   120   330   792  1716  3432  6427

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A000580.

Table[Sum[(-1)^r*Binomial[n, r]*Binomial[n - 8*r + k - 1, n - 1], {r, 0, Floor[k/8]}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, sum(r=0, floor(k/8), (-1)^r*binomial(n,r)*binomial(n-8*r + k-1,n-1))), ", "))) \\ G. C. Greubel, Nov 25 2017

T(n,k) = Sum_{r=0..floor(k/8)} (-1)^r*C(n,r)*C(n-8*r+k-1, n-1).

T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n > 1, T(n,3)=A000292(n) for n >= 3, T(n,4)=A000332(n) for n >= 7, T(n,5)=A000389(n) for n >= 9, T(n,6)=A000579(n) for n >= 11, T(n,7)=A000580(n) for n >= 13.

A378405 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+k,n-2*k).

Original entry on oeis.org

1, 1, 3, 13, 47, 171, 651, 2507, 9703, 37831, 148393, 584673, 2312267, 9174179, 36500257, 145566333, 581746503, 2329206823, 9341025429, 37516150599, 150874376997, 607479424817, 2448608334087, 9879562243961, 39897969991075, 161260133795371

Offset: 0

Seiichi Manyama, Nov 25 2024

nonn

Cf. A005725, A378406.

Cf. A001850, A082759.

Table[Sum[Binomial[n, k]*Binomial[n + k, n - 2*k], {k, 0, n/2}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 25 2024 *)

a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n+k, n-2*k));

a(n) = [x^n] (1 + x + x^2 * (1 + x)^2)^n.

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 4.1236218756427610008124277125077732535524468472302771364162049292... is the greatest root of the equation 31 - 256*d + 30*d^2 - 104*d^3 + 27*d^4 = 0 and c = 0.3580097056143148810957018324419771260252367819271313726816... is the positive real root of the equation -31 - 3024*c + 51376*c^2 - 257536*c^3 + 386304*c^4 = 0. - Vaclav Kotesovec, Nov 25 2024

A378406 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+2k,n-2k).

Original entry on oeis.org

1, 1, 3, 16, 67, 266, 1116, 4803, 20707, 89665, 390868, 1712283, 7527664, 33196606, 146800811, 650724896, 2890442051, 12862496583, 57331583055, 255915024714, 1143845768892, 5118643987872, 22930389117771, 102824420890590, 461502269341936, 2073064313021416

Offset: 0

Seiichi Manyama, Nov 25 2024

nonn

Cf. A005725, A378405.

a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n+2*k, n-2*k));

a(n) = [x^n] (1 + x + x^2 * (1 + x)^3)^n.

A104311 Numbers n such that the coefficient of x^n in (x^3+x^2+x+1)^n is prime.

Original entry on oeis.org

2, 4, 5, 8, 73, 649

Offset: 1

T. D. Noe, Mar 01 2005

n such that A005725(n) is prime. No other n<16000. The primes are in A104312. Only coefficients of the x, x^n, x^(2n) and x^(3n-1) terms can be prime; the coefficients of x and x^(3n-1) terms are prime whenever n is prime.

Any further terms are > 500000. - Lucas A. Brown, Oct 04 2024

Cf. A005725 (quadrinomial coefficients).

f=1; Do[f=Expand[f*(x^3+x^2+x+1)]; s=Coefficient[f, x, n]; If[PrimeQ[s], Print[{n, s}]], {n, 1000}]

cp's OEIS Frontend

A104312 Prime coefficient of x^n in (x^3+x^2+x+1)^n for n in A104311.

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A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.

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A378405 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+k,n-2*k).

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A378406 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+2*k,n-2*k).

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A104311 Numbers n such that the coefficient of x^n in (x^3+x^2+x+1)^n is prime.

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Mathematica

A378406 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+2k,n-2k).