A290000 -id:A290000 - cp's OEIS frontend

A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

1, 3, 1, 27, 12, 1, 729, 351, 39, 1, 59049, 29160, 3510, 120, 1, 14348907, 7144929, 882090, 32670, 363, 1, 10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1, 22876792454961, 11433166050879, 1427185336941, 54665851779, 674887059, 2685501, 3279, 1

Offset: 0

Roger L. Bagula, Feb 07 2010

Triangle T(n,k), read by rows, given by [3,6,27,72,243,702,2187,6480,...] DELTA [1,0,3,0,9,0,27,0,81,0,243,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

			Triangle begins as:
            1;
            3,          1;
           27,         12,         1;
          729,        351,        39,        1;
        59049,      29160,      3510,      120,      1;
     14348907,    7144929,    882090,    32670,    363,    1;
  10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), this sequence (q=3), A173008 (q=4).

Cf. A203148, A290000.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,3): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

(* First program *)
p[x_, n_, q_] = If[n==0, 1, Product[x + q^i, {i, 1, n}]];
Table[CoefficientList[p[x, n, 3], x], {n, 0, 10}] (* modified by G. C. Greubel, Feb 20 2021 *)
(* Second program *)
T[n_, k_, q_]:= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

p(x,n,q) = 1 if n=0, Product_{i=1..n} (x + q^i) otherwise, with q=3.
T(n,k) = 3^n*T(n-1,k) + T(n-1,k-1), T(0,0)=1. - Philippe Deléham, Oct 01 2011
Sum_{k=0..n} T(n, k, 4) = A290000(n+1). - G. C. Greubel, Feb 20 2021

Edited by G. C. Greubel, Feb 20 2021

A309327 a(n) = Product_{k=1..n-1} (4^k + 1).

Original entry on oeis.org

1, 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, 1678532740564688125136703125, 1760070825503098980191468752703125, 7382273863761775568111978346806480703125, 123854010565759745011512941023673583762640703125

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A027637, A052539, A053763.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), A290000 (m=3), this sequence (m=4).

[n lt 2 select 1 else (&*[4^j +1: j in [1..n-1]]): n in [0..15]]; // G. C. Greubel, Feb 21 2021

Table[Product[4^k + 1, {k, 1, n - 1}], {n, 0, 13}]
Join[{1}, Table[4^(Binomial[n,2])*QPochhammer[-1/4, 1/4, n-1], {n,15}]] (* G. C. Greubel, Feb 21 2021 *)

a(n) = prod(k=1, n-1, 4^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[4^(binomial(n,2))*q_pochhammer(n-1, -1/4, 1/4) for n in (1..15)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(4*x) / (1 - x).
G.f.: Sum_{k>=0} 2^(k*(k - 1)) * x^k / Product_{j=0..k-1} (1 - 4^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 4^k * a(k).
a(n) ~ c * 2^(n*(n - 1)), where c = Product_{k>=1} (1 + 1/4^k) = 1.355909673863479380345544...
a(n) = 4^(binomial(n+1,2))*(-1/4; 1/4){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

A119600 a(n) = 4*Product_{i=1..n-1} (3^i+1)^2.

Original entry on oeis.org

4, 4, 64, 6400, 5017600, 33738342400, 2008645953126400, 1070407428421058560000, 5124408580006984170864640000, 220656234047362257307900743516160000, 85495432669493277396354169745064287272960000, 298114237913837782686540845369489025952802406400000000, 9355246290649672947599943358541996936410690283965618585600000000

Offset: 0

N. J. A. Sloane, Jun 04 2006

nonn

R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des Nombres de Bordeaux, 14 (2002), 73-85.

Cf. A028362, A290000.

Table[4*Product[(3^i+1)^2,{i,n-1}],{n,0,12}] (* James C. McMahon, Sep 17 2024 *)

a(n) = 4*prod(i=1, n-1, (3^i+1)^2); \\ Michel Marcus, Oct 28 2015

a(n) = 4*A290000(n)^2. - Vaclav Kotesovec, Sep 17 2024

cp's OEIS Frontend

A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

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A309327 a(n) = Product_{k=1..n-1} (4^k + 1).

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A119600 a(n) = 4*Product_{i=1..n-1} (3^i+1)^2.

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