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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A159855 Riordan array ((1-2*x-x^2)/(1-x), x/(1-x)).

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -2, -2, 1, 1, -2, -4, -1, 2, 1, -2, -6, -5, 1, 3, 1, -2, -8, -11, -4, 4, 4, 1, -2, -10, -19, -15, 0, 8, 5, 1, -2, -12, -29, -34, -15, 8, 13, 6, 1, -2, -14, -41, -63, -49, -7, 21, 19, 7, 1, -2, -16, -55, -104, -112, -56, 14, 40, 26, 8, 1
Offset: 0

Views

Author

Philippe Deléham, Apr 24 2009

Keywords

Comments

The matrix inverse starts
1;
1, 1;
2, 0, 1;
2, 2, -1, 1;
4, 0, 3, -2, 1;
4, 4, -3, 5, -3, 1;
8, 0, 7, -8, 8, -4, 1;
8, 8, -7, 15, -16, 12, -5, 1;
16, 0, 15, -22, 31, -28, 17, -6, 1;
16, 16, -15, 37, -53, 59, -45, 23, -7, 1;
32, 0, 31, -52, 90, -112, 104, -68, 30, -8, 1;
- R. J. Mathar, Mar 29 2013

Examples

			Triangle begins:
   1;
  -1,  1;
  -2,  0,  1;
  -2, -2,  1,  1;
  -2, -4, -1,  2,  1;
  -2, -6, -5,  1,  3,  1;

Links

Crossrefs

Cf. A159854.

Programs

  • GAP
    Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)-2*Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)))); # Muniru A Asiru, Mar 22 2018
  • Magma
    /* As triangle */ [[Binomial(n, n-k)-2*Binomial(n-1, n-k-1)-Binomial(n-2, n-k-2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Mar 22 2018
  • Maple
    C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
  seq(C(n, n-k)-2*C(n-1, n-k-1)-C(n-2, n-k-2), k = 0..n)
end do; # Peter Bala, Mar 20 2018
  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - 2 # - #^2)/(1 - #)&, #/(1 - #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
  • Sage
    # uses[riordan_array from A256893]
riordan_array((1-2*x-x^2)/(1-x), x/(1-x), 8) # Peter Luschny, Mar 21 2018

Formula

T(n,k) = C(n,n-k) - 2*C(n-1,n-k-1) - C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

Extensions

Two data values in row 10 corrected by Peter Bala, Mar 20 2018

A370516 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^i for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, -5, 1, 7, -4, 1, -3, 3, -3, 1, 0, 0, 0, -2, 1, 0, 0, 0, -2, -1, 1, 0, 0, 0, -2, -3, 0, 1, 0, 0, 0, -2, -5, -3, 1, 1, 0, 0, 0, -2, -7, -8, -2, 2, 1, 0, 0, 0, -2, -9, -15, -10, 0, 3, 1, 0, 0, 0, -2, -11, -24, -25, -10, 3, 4
Offset: 0

Views

Author

Igor Victorovich Statsenko, Feb 21 2024

Keywords

Comments

Generalized binomial coefficients of the second order.

Examples

			n\k   0    1    2    3    4    5    6
0:    1
1:   -5    1
2:    7   -4    1
3:   -3    3   -3    1
4:    0    0    0   -2    1
5:    0    0    0   -2   -1    1
6:    0    0    0   -2   -3    0    1

Links

Crossrefs

For m=0 the formula gives the sequence A007318; for m=1 the formula gives the sequence A159854. In this case, we assume that A007318 consists of generalized binomial coefficients of order zero and A159854 consists of generalized binomial coefficients of order one.

Programs

  • Maple
    C:=(n,k)->n!/(k!*(n-k)!) : T:=(m,n,k)->sum(C(n+1,n-k-r)*Stirling2(r+m+1,r+1)*((-1)^r), r=0..n-k) : m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);

Formula

T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i where m = 2 for n >= 0, 0 <= k <= n.

A380851 Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.

Original entry on oeis.org

1, -2, 4, -2, 8, 16, -4, 24, 96, 64, -10, 80, 480, 640, 256, -28, 280, 2240, 4480, 3584, 1024, -84, 1008, 10080, 26880, 32256, 18432, 4096, -264, 3696, 44352, 147840, 236544, 202752, 90112, 16384, -858, 13728, 192192, 768768, 1537536, 1757184, 1171456, 425984, 65536
Offset: 0

Views

Author

Igor Victorovich Statsenko, Feb 06 2025

Keywords

Examples

			Triangle starts:
       k = 0      1       2        3        4        5       6
  n=0:     1;
  n=1:    -2,     4;
  n=2:    -2,     8,     16;
  n=3:    -4,    24,     96,      64;
  n=4:   -10,    80,    480,     640,     256;
  n=5:   -28,   280,   2240,    4480,    3584,    1024;
  n=6:   -84,  1008,  10080,   26880,   32256,   18432,   4096;

Links

Crossrefs

Columns: A002420 (k=0); A240530 (k=1).
Triangle for m=-3, r=1: A104713; for m=-2, r=1: A104712; for m=-1, r=1: A135278; for m=0, r=1: A007318; for m=1, r=1: A097805; for m=2, r=1: A159854.

Programs

  • Maple
    T:=(m,r,n,k)->add(binomial(i+m,m)*binomial(n+1,n-k-i)*r^(2*n)*(-1)^(i),i=0..n-k): m:=3/2: r:=2: seq(print(seq(T(m,r,n,k), k=0..n)), n=0..10);
  • Mathematica
    T[n_, k_] := 4^n Binomial[n, k] Hypergeometric2F1[3/2, k - n, k + 1, 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* Peter Luschny, Feb 07 2025 *)
  • SageMath
    # Using function riordan_array from A256893.
RA = riordan_array((1 - x)^(3/2 - 1), x/(1-x), 7)
for n in range(7): print(4^n * RA.row(n)[:n+1])  # Peter Luschny, Feb 28 2025

Formula

T(n,k) = Sum_{i=0..n-k} binomial(i+m, m)*binomial(n+1, n-k-i)*r^(2*n)*(-1)^(i), for m = 3/2 and r = 2.
From Peter Luschny, Feb 07 2025: (Start)
T(n,k) = r^(2*n)*JacobiP(n - k, 1 + k, m - 1 - n, -1).
T(n,k) = 4^n*binomial(n, k)*hypergeom([3/2, k - n], [k + 1], 1). (End)
Showing 1-3 of 3 results.

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