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.

A155856 Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 6, 1, 24, 42, 30, 10, 1, 120, 216, 168, 70, 15, 1, 720, 1320, 1080, 504, 140, 21, 1, 5040, 9360, 7920, 3960, 1260, 252, 28, 1, 40320, 75600, 65520, 34320, 11880, 2772, 420, 36, 1, 362880, 685440, 604800, 327600, 120120, 30888, 5544, 660, 45, 1
Offset: 0

Views

Author

Paul Barry, Jan 29 2009

Keywords

Comments

Row sums of B^{-1}*A155856*B^{-1} are A000166 with B=A007318.
Downward diagonals T(n+j, n) = j!*binomial(n+j, n) = j!*seq(j), where seq(j) are sequences A010965, A010967, ..., A011101, A017714, A017716, ..., A017764, for 6 <= j <= 50, respectively. - G. C. Greubel, Jun 04 2021

Examples

			Triangle begins:
     1;
     1,    1;
     2,    3,    1;
     6,   10,    6,    1;
    24,   42,   30,   10,    1;
   120,  216,  168,   70,   15,   1;
   720, 1320, 1080,  504,  140,  21,  1;
  5040, 9360, 7920, 3960, 1260, 252, 28, 1;

Links

Crossrefs

Cf. A155857 (row sums), A155858 (diagonal sums).

Programs

  • Mathematica
    Table[Binomial[2n-k,k](n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 24 2017 *)
  • Sage
    flatten([[factorial(n-k)*binomial(2*n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n,k) = binomial(2*n-k, k)*(n-k)!.
Sum_{k=0..n} T(n, k) = A155857(n)
Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
From G. C. Greubel, Jun 04 2021: (Start)
T(n, 0) = A000142(n). T(n+1, n) = A000217(n+1).
T(n+1, 1) = A007680(n). T(n+2, n) = A034827(n+4).
T(n+2, 2) = A175925(n). T(n+3, n) = A253946(n).
T(2*n, n) = A064352(n) T(n+4, n) = 4!*A000581(n).
T(n+1, n) = A000217(n+1). T(n+5, n) = 5!*A001287(n). (End)

A154987 Triangle read by rows: T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n!/k!)*(2*(n + k) - 1).

Original entry on oeis.org

-2, 4, 4, 13, 20, 13, 41, 69, 69, 41, 183, 268, 264, 268, 183, 1099, 1405, 1080, 1080, 1405, 1099, 7943, 9486, 5970, 4080, 5970, 9486, 7943, 65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547, 604831, 685672, 384552, 149520, 77280, 149520, 384552, 685672, 604831
Offset: 0

Views

Author

Roger L. Bagula, Jan 18 2009

Keywords

Examples

			     -2;
      4,     4;
     13,    20,    13;
     41,    69,    69,    41;
    183,   268,   264,   268,   183;
   1099,  1405,  1080,  1080,  1405,  1099;
   7943,  9486,  5970,  4080,  5970,  9486,  7943;
  65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547;
  ...

Links

Programs

  • Maple
    t:= proc(n,k) option remember; ## simplified t;
2*(n+k-1/2)*(n!/k!);
end proc:
A154987:= proc(n,k) ## n >= 0 and k = 0 .. n
t(n,k) + t(n,n-k)
end proc: # Yu-Sheng Chang, Apr 13 2020
  • Mathematica
    (* First program *)
t[n_, k_]:= 2*n!*Gamma[n+k+1/2]/(k!*Gamma[n+k-1/2]);
T[n_, k_]:= t[n, k] + t[n,n-k];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
(* Second Program *)
T[n_, k_]:= Binomial[n, k]*((n-k)!*(2*n+2*k-1) + k!*(4*n-2*k-1));
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 28 2020 *)
  • Sage
    def T(n, k): return binomial(n, k)*(factorial(n-k)*(2*n+2*k-1) + factorial(k)*(4*n-2*k-1))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 28 2020

Formula

T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*n!*Gamma(n + k + 1/2)/(k!*Gamma(n + k - 1/2)).
T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n+k-1/2)*(n!/k!). - Yu-Sheng Chang, Apr 13 2020
From G. C. Greubel, May 28 2020: (Start)
T(n,k) = binomial(n,k)*( (2*n+2*k-1)*(n-k)! + (4*n-2*k-1)*k! ).
T(n,n-k) = T(n,k), for k >= 0.
Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*n!*e_{n-1}(1) ), where e_{n}(x) is the finite exponential function = Sum_{k=0..n} x^k/k!.
Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*A007526(n) ).
T(n,0) = A175925(n-1) + 2*n.
T(n,1) = A007680(n) + A001107(n). (End)

Extensions

Partially edited by Andrew Howroyd, Mar 26 2020
Additionally edited by G. C. Greubel, May 28 2020

A364817 Triangle read by rows: T(n,k) = number of permutations p of [n] such that max(|p(p(i)) - i|)=k (n>=1, 0<=k<=n-1).

Original entry on oeis.org

1, 2, 0, 4, 0, 2, 10, 2, 6, 6, 26, 6, 22, 36, 30, 76, 24, 92, 144, 216, 168, 232, 80, 334, 640, 1150, 1524, 1080, 764, 312, 1328, 2984, 5516, 9712, 11784, 7920, 2620, 1152, 5234, 13296, 27668, 55750, 90240, 101400, 65520, 9496, 4616, 21780, 62124, 144564, 306272, 601756, 909312, 964080, 604800
Offset: 1

Views

Author

Seiichi Manyama, Oct 21 2023

Keywords

Examples

			Triangle starts:
     1;
     2,    0;
     4,    0,    2;
    10,    2,    6,     6;
    26,    6,   22,    36,    30;
    76,   24,   92,   144,   216,   168;
   232,   80,  334,   640,  1150,  1524,  1080;
   764,  312, 1328,  2984,  5516,  9712, 11784,   7920;
  2620, 1152, 5234, 13296, 27668, 55750, 90240, 101400, 65520;

Crossrefs

Columns k=0..1 give: A000085, A364819.
Row sums give A000142.
Cf. A130152, A175925.

Formula

T(n,n-1) = (2*n-5)*(n-2)! for n>3.
Showing 1-3 of 3 results.

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