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.

Previous Showing 11-14 of 14 results.

A113070 Expansion of ((1+x)/(1-2x))^2.

Original entry on oeis.org

1, 6, 21, 60, 156, 384, 912, 2112, 4800, 10752, 23808, 52224, 113664, 245760, 528384, 1130496, 2408448, 5111808, 10813440, 22806528, 47972352, 100663296, 210763776, 440401920, 918552576, 1912602624, 3976200192, 8254390272, 17112760320
Offset: 0

Views

Author

Paul Barry, Oct 14 2005

Keywords

Comments

Binomial transform is A014915. In general, ((1+x)/(1-r*x))^2 expands to a(n) = ((r+1)*r^n*((r+1)*n + r - 1) + 0^n)/r^2, which is also a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..k} (j+1)*(r+1)^j. This is the self-convolution of the coordination sequence for the infinite tree with valency r.

Links

Crossrefs

Cf. A113071.

Programs

  • Magma
    [3*2^n*(3*n+1)/4+0^n/4: n in [0..30]]; // Vincenzo Librandi, May 21 2011
  • Mathematica
    Join[{1},LinearRecurrence[{4,-4},{6,21},30]] (* or *) CoefficientList[ Series[((1+x)/(1-2x))^2,{x,0,30}],x] (* Harvey P. Dale, May 20 2011 *)

Formula

G.f.: (1+x)^2/(1-2x)^2;
a(n) = 3*2^n(3n+1)/4 + 0^n/4;
a(n) = Sum_{k=0..n} A003945(k)*A003945(n-k);
a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*3^j.
a(n) = 4*a(n-1) - 4*a(n-2); a(0)=1, a(1)=6, a(2)=21. - Harvey P. Dale, May 20 2011

A119673 T(n, k) = 3*T(n-1, k-1) + T(n-1, k) for k < n and T(n, n) = 1, T(n, k) = 0, if k < 0 or k > n; triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 13, 1, 1, 10, 34, 40, 1, 1, 13, 64, 142, 121, 1, 1, 16, 103, 334, 547, 364, 1, 1, 19, 151, 643, 1549, 2005, 1093, 1, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 1, 28, 349, 2542, 11926, 37384, 78322, 105796, 83653, 29524, 1
Offset: 0

Views

Author

Zerinvary Lajos, Jun 11 2006

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1,  7,  13,    1;
  1, 10,  34,   40,    1;
  1, 13,  64,  142,  121,     1;
  1, 16, 103,  334,  547,   364,     1;
  1, 19, 151,  643, 1549,  2005,  1093,     1;
  1, 22, 208, 1096, 3478,  6652,  7108,  3280,    1;
  1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1;

Links

Crossrefs

Cf. A003462, A014915, A081271, A119258.

Programs

  • Magma
    function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return 3*T(n-1,k-1) + T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019
  • Maple
    T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)* hypergeom([1,n+1],[k+2],m)/(k+1)!; A119673 := (n,k) -> T(n,k,3);
seq(print(seq(round(evalf(A119673(n,k))),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014
  • Mathematica
    T[, 0]=1; T[n, n_]=1; T[n_, k_]/; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
  • PARI
    T(n,k) = if(k<0 || k>n, 0, if(k==n, 1, 3*T(n-1, k-1) +T(n-1,k)));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 18 2019
  • Sage
    @CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return 3*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

Formula

T(n,k) = R(n,k,3) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k, k+1)* hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

Extensions

Definition clarified by Philippe Deléham, Jun 13 2006
Entry revised by N. J. A. Sloane, Jun 19 2006

A246797 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-2)^k.

Original entry on oeis.org

1, 5, 2, 17, 14, 3, 49, 62, 27, 4, 129, 222, 147, 44, 5, 321, 702, 627, 284, 65, 6, 769, 2046, 2307, 1404, 485, 90, 7, 1793, 5630, 7683, 5884, 2725, 762, 119, 8, 4097, 14846, 23811, 22012, 12805, 4794, 1127, 152, 9, 9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10
Offset: 0

Views

Author

Derek Orr, Nov 15 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-2)^0 + A_1*(x-2)^1 + A_2*(x-2)^2 + ... + A_n*(x-2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			Triangle starts:
1;
5,        2;
17,      14,     3;
49,      62,    27,     4;
129,    222,   147,    44,     5;
321,    702,   627,   284,    65,     6;
769,   2046,  2307,  1404,   485,    90,    7;
1793,  5630,  7683,  5884,  2725,   762,  119,    8;
4097, 14846, 23811, 22012, 12805,  4794, 1127,  152,   9;
9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10;
...

Crossrefs

Cf. A246788, A014106, A000337, A246799.

Programs

  • PARI
    T(n,k) = (k+1)*sum(i=0,n-k,2^i*binomial(i+k+1,k+1))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

Formula

T(n,0) = n*2^(n+1)+1, for n >= 0.
T(n,n-1) = n*(2*n+3), for n >= 1.
Row n sums to A014915(n-1) = T(n,0) of A246799.

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3)^k.

Original entry on oeis.org

1, 7, 2, 34, 20, 3, 142, 128, 39, 4, 547, 668, 309, 64, 5, 2005, 3098, 1929, 604, 95, 6, 7108, 13304, 10434, 4384, 1040, 132, 7, 24604, 54128, 51258, 27064, 8600, 1644, 175, 8, 83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9, 280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10
Offset: 0

Views

Author

Derek Orr, Nov 15 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-3)^0 + A_1*(x-3)^1 + A_2*(x-3)^2 + ... + A_n*(x-3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			Triangle starts:
1;
7,           2;
34,         20,       3;
142,       128,      39,      4;
547,       668,     309,     64,      5;
2005,     3098,    1929,    604,     95,      6;
7108,    13304,   10434,   4384,   1040,    132,     7;
24604,   54128,   51258,  27064,   8600,   1644,   175,    8;
83653,  211592,  234966, 149536,  59630,  15252,  2443,  224,   9;
280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10;
...

Crossrefs

Cf. A246797, A014915, A140676, A246788.

Programs

  • PARI
    T(n, k) = (k+1)*sum(i=0, n-k, 3^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,0) = ((2*n+1)*3^(n+1) + 1)/4, for n >= 0.
T(n,n-1) = n*(3*n+4), for n >= 1.
Row n sums to A014916(n+1) = T(2*n+1,0) of A246788.
Previous Showing 11-14 of 14 results.

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

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