cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A118180 Triangle T(n, k) = 3^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 27, 81, 27, 1, 1, 81, 729, 729, 81, 1, 1, 243, 6561, 19683, 6561, 243, 1, 1, 729, 59049, 531441, 531441, 59049, 729, 1, 1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1, 1, 6561, 4782969, 387420489, 3486784401, 3486784401, 387420489, 4782969, 6561, 1
Offset: 0

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Author

Paul D. Hanna, Apr 15 2006

Keywords

Comments

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-3^n*x).

Examples

			A(x,y) = 1/(1-xy) + x/(1-3xy) + x^2/(1-9xy) + x^3/(1-27xy) + ...
Triangle begins:
  1;
  1,    1;
  1,    3,      1;
  1,    9,      9,        1;
  1,   27,     81,       27,        1;
  1,   81,    729,      729,       81,        1;
  1,  243,   6561,    19683,     6561,      243,      1;
  1,  729,  59049,   531441,   531441,    59049,    729,    1;
  1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1; ...
The matrix inverse T^-1 starts:
      1;
     -1,     1;
      2,    -3,     1;
    -10,    18,    -9,    1;
    134,  -270,   162,  -27,   1;
  -4942, 10854, -7290, 1458, -81, 1; ...
where [T^-1](n,k) = A118183(n-k)*(3^k)^(n-k).

Links

Crossrefs

Cf. A118181 (row sums), A118182 (antidiagonal sums), A118183, A118184.
Cf. A117401 = ConvOffsStoT transform of 2^n.
Cf. A117401 (m=0), this sequence (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

Programs

  • Magma
    A118180:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A118180(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021
  • Maple
    seq(seq( (3^k)^(n-k), k=0..n), n=0..12);
  • Mathematica
    T[n_, k_, m_]:= (m+2)^(k*(n-k)); Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)
  • PARI
    T(n,k) = if(k<0 || k>n, 0, 3^(k*(n-k)));
  • Sage
    def A118180(n, k, m): return (m+2)^(k*(n-k))
flatten([[A118180(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Formula

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-3^n*x*y). G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,3*y).
Equals ConvOffsStoT transform of the 3^n series: (1, 3, 9, 27, ...); e.g., ConvOffs transform of (1, 3, 9, 27) = (1, 27, 81, 27, 1). - Gary W. Adamson, Apr 21 2008
T(n,k) = (1/n)*( 3^(n-k)*k*T(n-1,k-1) + 3^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 1. - G. C. Greubel, Jun 28 2021

A118184 Column 0 of the matrix log of triangle A118180, after term in row n is multiplied by n: a(n) = n*[log(A118180)](n,0), where A118180(n,k) = 3^(k*(n-k)).

Original entry on oeis.org

0, 1, -1, 3, -23, 329, 18231, -22030373, 34718491601, -130548608723439, 1300095260497408879, -35497483240662990289357, 2687397326811421691366217657, -562747611676887059779727492799911, 320110532506391993959111359699070808231
Offset: 0

Views

Author

Paul D. Hanna, Apr 15 2006

Keywords

Comments

The entire matrix log of triangle A118180 is determined by column 0 (this sequence): [log(A118180)](n,k) = a(n-k)/(n-k)*(3^k)^(n-k) for n>k>=0.

Examples

			Column 0 of log(A118180) = [0, 1, -1/2, 3/3, -23/4, 329/5, 18231/6, ...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + ...
= x/(1-3*x) - x^2/(1-9*x) + 3*x^3/(1-27*x) - 23*x^4/(1-81*x) + 329*x^5/(1-243*x) + 18231*x^6/(1-729*x) - 22030373*x^7/(1-2187*x) + ...

Links

Crossrefs

Cf. A118180, A118183.

Programs

  • Mathematica
    A118183[n_]:= A118183[n]= If[n<2, (-1)^n, -Sum[3^(j*(n-j))*A118183[j], {j,0,n-1}]];
a[n_]:= a[n]= -Sum[3^(j*(n-j))*j*A118183[j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)
  • PARI
    {a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(3^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}
  • Sage
    @CachedFunction
def A118183(n): return (-1)^n if (n<2) else -sum(3^(j*(n-j))*A118183(j) for j in (0..n-1))
def a(n): return (-1)*sum( 3^(j*(n-j))*j*A118183(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Formula

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-3^n*x).
By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118183(n-k)*(3^k)^(n-k) for n>=0.
a(3^n) is divisible by 3^n.

A118181 Row sums of triangle A118180: a(n) = Sum_{k=0..n} (3^k)^(n-k) for n>=0.

Original entry on oeis.org

1, 2, 5, 20, 137, 1622, 33293, 1182440, 72811793, 7757988842, 1433154521621, 458101483131260, 253879024041595289, 243453910296759945662, 404765167247068325944349, 1164432505878183620543030480
Offset: 0

Views

Author

Paul D. Hanna, Apr 15 2006

Keywords

Comments

Also equals column 0 of the matrix square of triangle A118180, where [A118180^2](n,k) = a(n-k)*(3^k)^(n-k) for n>=k>=0.

Examples

			A(x) = 1/(1-x) + x/(1-3x) + x^2/(1-9x) + x^3/(1-27x) + ...
= 1 + 2*x + 5*x^2 + 20*x^3 + 137*x^4 + 1622*x^5 + 33293*x^6 +...

Links

Crossrefs

Cf. A118180 (triangle), A118182 (antidiagonal sums); A118183, A118184.

Programs

  • Magma
    [(&+[3^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
  • Maple
    seq( add(3^(k*(n-k)), k=0..n), n=0..30); # modified by G. C. Greubel, Jun 29 2021
  • Mathematica
    Table[Sum[3^(k*(n-k)), {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)
  • PARI
    a(n)=sum(k=0, n, (3^k)^(n-k) );
  • Sage
    [sum(3^(k*(n-k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Formula

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

A118182 Antidiagonal sums of triangle A118180: a(n) = Sum_{k=0..[n/2]} (3^k)^(n-2*k) for n>=0.

Original entry on oeis.org

1, 1, 2, 4, 11, 37, 164, 1000, 8021, 81001, 1076006, 19683244, 473632031, 14349084877, 571833704648, 31381448626000, 2265367321680041, 205893684435186001, 24615565942378859210, 4052605390737766057684
Offset: 0

Views

Author

Paul D. Hanna, Apr 15 2006

Keywords

Examples

			A(x) = 1/(1-x^2) + x/(1-3x^2) + x^2/(1-9x^2) + x^3/(1-27x^2) +...
= 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 164*x^6 + 1000*x^7 +...

Links

Crossrefs

Cf. A118180 (triangle), A118181 (row sums), A118183, A118184.

Programs

  • Magma
    [(&+[3^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
  • Mathematica
    Table[Sum[3^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)
  • PARI
    a(n)=sum(k=0, n\2, (3^k)^(n-2*k) );
  • Sage
    [sum(3^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Formula

G.f.: A(x) = Sum_{n>=0} x^n/(1-3^n*x^2).
a(2*n) = Sum_{k=0..n} (3^k)^(2*(n-k)).
a(2*n+1) = Sum_{k=0..n} (3^k)^(2*(n-k) +1).
Showing 1-4 of 4 results.

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