A173007 -id:A173007 - cp's OEIS frontend

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins:
      1;
      2,     1;
      8,     6,    1;
     64,    56,   14,    1;
   1024,   960,  280,   30,  1;
  32768, 31744, 9920, 1240, 62, 1;

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

Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).

Cf. A006125, A022166, A028362.

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

T[n_, k_, q_]:= 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,2], {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,2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019

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.

Original entry on oeis.org

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

A203148 (n-1)-st elementary symmetric function of {3,9,...,3^n}.

Original entry on oeis.org

1, 12, 351, 29160, 7144929, 5223002148, 11433166050879, 75035879252272080, 1477081305957768349761, 87223128348206814118735932, 15451489966710801620870785316511, 8211586182553137756809552940033725880, 13091937140529934508508023103481190655434529

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the integers 3^j, j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
  1;
  1   3;
  1  12    27;
  1  39   351    729;
  1 120  3510  29160   59049;
  1 363 32670 882090 7144929 14348907;
which is the row-reversed version of A173007. This here is the first subdiagonal. The diagonal seems to be A047656. The first column is A029858. (End)

G. C. Greubel, Table of n, a(n) for n = 1..60

Cf. A203149.

Cf. A029858, A047656, A173007.

[(1/2)*(3^n -1)*3^(Binomial(n,2)): n in [1..20]]; // G. C. Greubel, Feb 24 2021

f[k_]:= 3^k; t[n_]:= Table[f[k], {k, 1, n}];
a[n_]:= SymmetricPolynomial[n - 1, t[n]];
Table[a[n], {n, 1, 16}] (* A203148 *)
Table[1/2 (3^n - 1) 3^Binomial[n, 2], {n, 1, 20}] (* Emanuele Munarini, Sep 14 2017 *)

[(1/2)*(3^n -1)*3^(binomial(n,2)) for n in (1..20)] # G. C. Greubel, Feb 24 2021

a(n) = (1/2)*(3^n-1)*3^(binomial(n,2)). - Emanuele Munarini, Sep 14 2017

A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 32, 139, 108, 1, 288, 8331, 35692, 27648, 1, 3413, 908331, 26070067, 111565148, 86400000, 1, 50069, 160145259, 42405161203, 1216436611100, 5205269945088, 4031078400000

Offset: 0

Seiichi Manyama, Sep 24 2021

			Triangle begins:
  1;
  1,    1;
  1,    5,      4;
  1,   32,    139,      108;
  1,  288,   8331,    35692,     27648;
  1, 3413, 908331, 26070067, 111565148, 86400000;

Seiichi Manyama, Rows n = 0..37, flattened

Column k=1 gives A001923.
The diagonal of the triangle is A002109.
Cf. A007318, A008955, A023531, A108084, A173007, A173008, A249677.

T(n, k) = if(k==0, 1, if(k==n, prod(j=1, n, j^j), T(n-1, k)+n^n*T(n-1, k-1)));

PARI
```
row(n) = Vecrev(prod(k=1, n, 1+k^k*x));
```

T(0,0) = 1; T(n,k) = T(n-1,k) + n^n * T(n-1,k-1).

cp's OEIS Frontend

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

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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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A203148 (n-1)-st elementary symmetric function of {3,9,...,3^n}.

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Programs

Magma

Mathematica

Sage

Formula

A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

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Programs

PARI

PARI

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