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 1-3 of 3 results.

A089513 Third column of triangle A089504.

Original entry on oeis.org

1, 90, 6156, 387720, 23705136, 1432922400, 86230132416, 5179923146880, 310942155338496, 18660051727004160, 1119687641441381376, 67183287394552842240, 4031045937469026349056, 241863924899255181189120
Offset: 0

Views

Author

Wolfdieter Lang, Dec 01 2003

Keywords

Comments

Convolution of A089507 with powers of 60.

Links

Crossrefs

Cf. A089505, A089507, A089514.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-90 x+1944 x^2-8640 x^3),{x,0,20}],x] (* or *) LinearRecurrence[{90,-1944,8640},{1,90,6156},20] (* Harvey P. Dale, Jul 07 2021 *)

Formula

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).
a(n)=A089504(n+3, 3), n>=0.
a(n)= (50*(5*4*3)^n - 24*(4*3*2)^n + (3*2*1)^n)/27 = (6^n)*(5*10^(n+1) - 3*2^(2*n+3) + 1)/27.

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Original entry on oeis.org

1, 30, 756, 18360, 441936, 10614240, 254788416, 6115201920, 146766525696, 3522406694400, 84537821131776, 2028908069959680, 48693795855814656, 1168651113600245760, 28047626804770062336, 673143043784666480640
Offset: 0

Views

Author

Wolfdieter Lang, Dec 01 2003

Keywords

Comments

Convolution of A000400 (powers of 6) with A009968 (powers of 24).

Links

Crossrefs

Cf. A089505, A089513, A089514.

Programs

  • Magma
    [6^n*(4^(n+1)-1)/3: n in [0..15]]; // Vincenzo Librandi, Oct 18 2017
  • Mathematica
    CoefficientList[Series[1/((1-6x)(1-24x)),{x,0,20}],x] (* or *) LinearRecurrence[{30,-144},{1,30},20] (* Harvey P. Dale, Sep 25 2017 *)

Formula

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)).
a(n) = A089504(n+2, 2), n>=0.
a(n) = (4*(4*3*2)^n - (3*2*1)^n)/3 = (2^n)*(2^(2*(n+1))-1)*3^(n-1).
a(n) = 6^n*(4^(n+1)-1)/3. - Vincenzo Librandi, Oct 18 2017

A089519 Fourth column (k=6) of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, 882, 186876, 28245672, 3762380016, 474431543712, 58322293189056, 7082435837377152, 854925864902090496, 102893307861680404992, 12365333752840511118336, 1484928368468173355231232
Offset: 0

Views

Author

Wolfdieter Lang, Dec 01 2003

Keywords

References

  • P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

Links

Crossrefs

Cf. A089514, A089517-8, A090219.

Formula

G.f.: (1+672*x+14400*x^2)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)*(1-6*5*4*x)).
a(n)= 20*(6*5*4)^n -30*(5*4*3)^n + 12*(4*3*2)^n - (3*2*1)^n = b(n) + 672*b(n-1) + 14400*b(n-2), with b(n) := A089514(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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