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 1-2 of 2 results.

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512
Offset: 0

Views

Author

Dale Gerdemann, Apr 12 2015

Keywords

Comments

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
a\b 1.......2.......3.......4.......5.......6
-1 A144431
0 A007318 A038208 A038221
1 A008292 A256890 A257180 A257606 A257607
2 A060187 A257609 A257611 A257613 A257615
3 A142458 A257610 A257620 A257622 A257624 A257626
4 A142459 A257612 A257621
5 A142460 A257614 A257623
6 A142461 A257616 A257625
7 A142462 A257617 A257627
8 A167884 A257618
9 A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
0 A000079 A000302 A000400
1 A000142 A001715 A001725 A049388 A049198
2 A000165 A002866 A002866 A051580 A051582
3 A008544 A051578 A037559 A051605 A051607 A051609
4 A001813 A047053 A000407 A034177 A051618 A051620 A051622
5 A047055 A008546 A008548 A034300 A034325 A051688 A051690
6 A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
7 A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
8 A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
9 A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
10 A051262 A035265 A035273 A035277
11 A254322
12 A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
a\b 0 1 2 3
-2 A130595/1
-1
0
1 A110555/-1 A120434/-1 A144697/1 A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

Examples

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Links

Crossrefs

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.
Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

Programs

  • Magma
    A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)
  • PARI
    t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015
  • SageMath
    def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

Formula

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A380865 Triangle read by rows: T(n, k) = 2^(2*n)*JacobiP(n - k, k, -1/2 - n, -1).

Original entry on oeis.org

1, 2, 4, 6, 24, 16, 20, 120, 160, 64, 70, 560, 1120, 896, 256, 252, 2520, 6720, 8064, 4608, 1024, 924, 11088, 36960, 59136, 50688, 22528, 4096, 3432, 48048, 192192, 384384, 439296, 292864, 106496, 16384, 12870, 205920, 960960, 2306304, 3294720, 2928640, 1597440, 491520, 65536
Offset: 0

Views

Author

Peter Luschny, Feb 07 2025

Keywords

Examples

			Triangle begins:
  [0]    1;
  [1]    2,     4;
  [2]    6,    24,     16;
  [3]   20,   120,    160,     64;
  [4]   70,   560,   1120,    896,    256;
  [5]  252,  2520,   6720,   8064,   4608,   1024;
  [6]  924, 11088,  36960,  59136,  50688,  22528,   4096;
  [7] 3432, 48048, 192192, 384384, 439296, 292864, 106496, 16384;

Crossrefs

Cf. A038234, A380851, A097807, A128908, A380864 (row sums).

Programs

  • Maple
    T := (n, k) -> 2^(2*n)*JacobiP(n - k, k, -1/2 - n, -1):
seq(print(seq(simplify(T(n, k)), k=0..n)), n=0..9);
  • Mathematica
    T[n_, k_] := 4^n Binomial[n, k] Hypergeometric2F1[1/2, k - n, k + 1, 1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten

Formula

Consider a family of Jacobi polynomials defined with a rational number r as
J(n, k, r, x) = denominator(r)^(2*n)*JacobiP(n - k, k, r - n, x).
For r = -1/2 and x = -1 is J(n, k, r, x) = T(n, k).
For r = 1/2 and x = -1 is J(n, k, r, x) = A380851(n, k).
For r = 1/2 or r = -1/2 and x = 1 is J(n, k, r, x) = A038234(n, k).
The choice r = n and x = -1 gives Riordan array A097807, (1/(1 + x), 1).
The choice r = k and x = -1 gives Riordan array A128908, (1, x/(1 - x)^2).
The choice r = n and x = 1 gives the Pascal triangle.
T(n, k) = 4^n*binomial(n, k)*hypergeom([1/2, k - n], [k + 1], 1).
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.