A092896 -id:A092896 - cp's OEIS frontend

A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.

1, 4, 1, 64, 20, 1, 4096, 1344, 84, 1, 1048576, 348160, 22848, 340, 1, 1073741824, 357564416, 23744512, 371008, 1364, 1, 4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1, 72057594037927936, 24017731997138944, 1600791219535872, 25384570585088, 99158478848, 95414592, 21844, 1

Offset: 0

Roger L. Bagula, Feb 07 2010

Row sums are 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, ... (partial products of A092896).

Triangle T(n,k), read by rows, given by [4,12,64,240,1024,4032,16384,...] DELTA [1,0,4,0,16,0,64,0,256,0,1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

			Triangle begins as:
              1;
              4,             1;
             64,            20,           1;
           4096,          1344,          84,          1;
        1048576,        348160,       22848,        340,       1;
     1073741824,     357564416,    23744512,     371008,    1364,    1;
  4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1;

Robert Israel, Table of n, a(n) for n = 0..1430

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), A173007 (q=3), this sequence (q=4).

Cf. A092896, A108084, A309327.

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,4): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

P:= 1: A:= 1:
for n from 1 to 12 do
  P:= expand(P*(x+4^n));
  A:= A, seq(coeff(P,x,j),j=0..n)
od:
A; # Robert Israel, Aug 12 2015

(* First program *)
p[x_, n_, q_]= If[n==0, 1, Product[x + q^i, {i,n}]];
Table[CoefficientList[p[x, n, 4], x], {n, 0, 10}]//Flatten (* 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,4], {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,4) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

T(n,k) = 4^n*T(n-1,k) + T(n-1,k-1) with T(0,0)=1. - Philippe Deléham, Oct 01 2011

Sum_{k=0..n} T(n, k, 4) = A309327(n+1). - G. C. Greubel, Feb 20 2021

A092897 Expansion of (1-x-x^2-3x^3) / ((1+x)^2(1-3*x)).

Original entry on oeis.org

1, 0, 4, 4, 24, 56, 188, 540, 1648, 4912, 14772, 44276, 132872, 398568, 1195756, 3587212, 10761696, 32285024, 96855140, 290565348, 871696120, 2615088280, 7845264924, 23535794684, 70607384144, 211822152336, 635466457108, 1906399371220, 5719198113768, 17157594341192

Offset: 0

Paul Barry, Mar 12 2004

Binomial transform is A092896.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,3).

Cf. A092896.

[(3^n +4*0^n -(-1)^n*(1-4*n))/4: n in [0..30]]; // G. C. Greubel, Feb 20 2021

LinearRecurrence[{1,5,3},{1,0,4,4},30] (* Harvey P. Dale, Mar 24 2018 *)

Vec((1 - x - x^2 - 3*x^3) / ((1 + x)^2 * (1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 25 2016

[(3^n +4*0^n -(-1)^n*(1-4*n))/4 for n in [0..30]]; # G. C. Greubel, Feb 20 2021

a(n) = (3^n + 4 * 0^n - (-1)^n + 4*n*(-1)^n)/4.
a(n) = a(n-1) + 5*a(n-2) + 3*a(n-3) for n>3. - Colin Barker, Nov 25 2016
E.g.f.: (4 +exp(3*x) -(1+4*x)*exp(-x))/4. - G. C. Greubel, Feb 20 2021

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).

Original entry on oeis.org

1, 0, 4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 0

Paul Barry, Mar 12 2004

Partial sums are A092896.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A002001, A092896, A110594.

[1,0,4] cat [3*4^(n-2): n in [3..30]]; // G. C. Greubel, Feb 21 2021

a:= n-> 3*4^n/16+13*0^n/16+add(binomial(n,k)*(-1)^k*(3*k/4+k*(k-1)/2), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2018

Join[{1, 0, 4}, LinearRecurrence[{4}, {12}, 22]] (* Jean-François Alcover, Sep 16 2019 *)

Vec((1 -4*x +4*x^2 -4*x^3)/(1-4*x) + O(x^30)) \\ Andrew Howroyd, Nov 03 2018

[1,0,4]+[3*4^(n-2) for n in (3..30)] # G. C. Greubel, Feb 21 2021

a(n+2) = 4 * A002001(n).
a(n) = (3*4^n + 13*0^n)/16 + Sum_{k=0..n} binomial(n, k)*(-1)^k*(3*k/4 + k*(k-1)/2).
G.f.: 1 - x + 8*x^2 + 2*x/G(0), where G(k) = 1 + 1/(1 - x*(3*k+4)/(x*(3*k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
a(n) = A110594(n-1) for n >= 2. - Georg Fischer, Nov 03 2018
From G. C. Greubel, Feb 21 2021: (Start)
a(n) = (3*4^n +16*[n=2] -12*[n=1] +13*0^n)/16.
E.g.f.: (13 -12*x + 8*x^2 + 3*exp(4*x))/16. (End)

cp's OEIS Frontend

A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.

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A092897 Expansion of (1-x-x^2-3*x^3) / ((1+x)^2*(1-3*x)).

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Magma

Mathematica

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Sage

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A092898 Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).

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Author

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Comments

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A092897 Expansion of (1-x-x^2-3x^3) / ((1+x)^2(1-3*x)).

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).