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.

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A249757 Triangular array of coefficients of polynomials p(n,x) = (x + 1)*p(n-1,x) + 2*n*x, p(0,x) = 1.

Original entry on oeis.org

1, 1, 3, 1, 8, 3, 1, 15, 11, 3, 1, 24, 26, 14, 3, 1, 35, 50, 40, 17, 3, 1, 48, 85, 90, 57, 20, 3, 1, 63, 133, 175, 147, 77, 23, 3, 1, 80, 196, 308, 322, 224, 100, 26, 3, 1, 99, 276, 504, 630, 546, 324, 126, 29, 3, 1, 120, 375, 780, 1134, 1176, 870, 450, 155
Offset: 0

Views

Author

Clark Kimberling, Nov 07 2014

Keywords

Comments

(Sum of numbers in row n) = A097809(n) for n >= 0.

Examples

			p(0,x) = 1
p(1,x) = 1 + 3*x
p(2,x) = 1 + 8*x + 3*x^2
First 6 rows:
1
1   3
1   8    3
1   15   11    3
1   24   26    14   3
1   35   50    40   17   3

Links

Crossrefs

Cf. A249755, A249756, A097809.

Programs

  • Mathematica
    z = 14; p[n_, x_] := (x + 1) p[n - 1, x] + 2*n*x; p[0, x_] = 1;
t = Table[Factor[p[n, x]], {n, 0, z}]
TableForm[Rest[Table[CoefficientList[t[[n]], x], {n, 0, z}]]] (* A249757 array *)
Flatten[CoefficientList[t, x]] (* A249757 sequence *)

A097810 a(n) = 7*2^n - 3*n - 6.

Original entry on oeis.org

1, 5, 16, 41, 94, 203, 424, 869, 1762, 3551, 7132, 14297, 28630, 57299, 114640, 229325, 458698, 917447, 1834948, 3669953, 7339966, 14679995, 29360056, 58720181, 117440434, 234880943, 469761964, 939524009, 1879048102, 3758096291
Offset: 0

Views

Author

Paul Barry, Aug 25 2004

Keywords

Links

Crossrefs

Cf. A079583, A097809, A131068.

Programs

  • Mathematica
    s=1;lst={s};Do[s+=(s+=n)+n++;AppendTo[lst, s], {n, 1, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[7*2^n-3n-6,{n,0,30}] (* or *) LinearRecurrence[{4,-5,2},{1,5,16},30] (* Harvey P. Dale, Nov 15 2011 *)

Formula

G.f.: (1 + x + x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 3*n, n > 0, a(0)=1.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
From Elmo R. Oliveira, Mar 06 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 3*(x + 2)).
a(n) = A131068(n+1)/2. (End)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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