A060722 -id:A060722 - cp's OEIS frontend

A251671 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^k.

1, 6, 180, 43398, 88701816, 1573206748746, 248688444559874580, 356335498302585834118638, 4663871943514788530035646937456, 558720685051192771669885091319459750546, 612058892657175926094223171960469926874935754700

Offset: 0

Paul D. Hanna, Jan 21 2015

nonn

			G.f.: A(x) = 1 + 6*x + 180*x^2 + 43398*x^3 + 88701816*x^4 + 1573206748746*x^5 +...
where A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3^2)*x^2/(1-x)^3 + (2^3+3^3)^3*x^3/(1-x)^4 +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^0 + 1*(2^1+3^1)^1 = 6;
a(2) = 1*(2^0+3^0)^0 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^2 = 180;
a(3) = 1*(2^0+3^0)^0 + 3*(2^1+3^1)^1 + 3*(2^2+3^2)^2 + 1*(2^3+3^3)^3 = 43398;
a(4) = 1*(2^0+3^0)^0 + 4*(2^1+3^1)^1 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^3 + 1*(2^4+3^4)^4 = 88701816; ...

Cf. A251661, A251184, A060722.

Table[Sum[Binomial[n,k] * (2^k + 3^k)^k,{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jan 25 2015 *)

{a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3^k)^k )}
for(n=0, 15, print1(a(n), ", "))

{a(n)=local(A=1); A=sum(m=0, n, (2^m + 3^m)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

G.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1-x)^(n+1).

a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jan 25 2015

A326377 For any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; a(n) = g(f(n) o f(n)) (where o denotes function composition).

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 29, 8, 81, 1100, 59, 48, 101, 195478444, 40425, 16, 157, 648, 229, 440000, 64240097649, 1445390468875226977004, 313, 192, 214358881, 44574662297516497591170630280506162081362246142404, 19683, 9921285858330292941824, 421, 72765000, 547, 32

Offset: 1

Rémy Sigrist, Jul 02 2019

nonn

This sequence is the main diagonal of A326376.

			The first terms, alongside the corresponding polynomials, are:
  n   a(n)  f(n)   f(n) o f(n)
  --  ----  -----  -----------
   1     1      0            0
   2     2      1            1
   3     3      x            x
   4     4      2            2
   5    11    x^2          x^4
   6    12    x+1          x+2
   7    29    x^3          x^9
   8     8      3            3
   9    81    2*x          4*x
  10  1100  x^2+1  x^4+2*x^2+2
  11    59    x^4         x^16
  12    48    x+2          x+4

Cf. A060722, A243896, A326376.

g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i, 2] * v^(primepi(f[i, 1]) - 1))
a(n) = g(f(n, f(n)))

a(n) = A326376(n, n).
a(2^k) = 2^k for any k >= 0.
a(3^k) = A060722(k) for any k >= 0.
a(prime(k)) = A243896(k) for any k >= 1 (where prime(k) denotes the k-th prime number).

A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 0, 3, 18, 24, 39, 3456, 8190, 5928, 2109, 7619508, 17094240, 13700700, 4215120, 417153, 149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179, 26394940582090344, 59388527912287392, 47325384827973252, 15262273318168800, 1648005959253654, 74268805562952, 1233891662727

Offset: 0

Geoffrey Critzer, Jul 16 2021

			             1;
             0,            3;
            18,           24,           39;
          3456,         8190,         5928,        2109;
       7619508,     17094240,     13700700,     4215120,     417153;
  149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).

nn = 7; q = 3; b[p_, i_] := Count[p, i]; d[p_, i_] :=Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Table[Take[(Table[ Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v]^3 Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],n], {n, 1, nn}] // Grid

cp's OEIS Frontend

A251671 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^k.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica