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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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A352650 Triangle read by rows: T(n,k) = n * T(n-1,k) + (-1)^(n-k) for 0 <= k <= n with initial values T(n,k) = 0 if n < 0 or k < 0 or k > n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 4, 2, 1, 9, 15, 9, 3, 1, 44, 76, 44, 16, 4, 1, 265, 455, 265, 95, 25, 5, 1, 1854, 3186, 1854, 666, 174, 36, 6, 1, 14833, 25487, 14833, 5327, 1393, 287, 49, 7, 1, 133496, 229384, 133496, 47944, 12536, 2584, 440, 64, 8, 1, 1334961, 2293839, 1334961, 479439, 125361, 25839, 4401, 639, 81, 9, 1
Offset: 0

Views

Author

Werner Schulte, Apr 04 2022

Keywords

Comments

Conjecture 1: T(n,k) = Sum_{i=0..n-k} (-1)^(n+k+i) * A326326(n-k,i) * n^i for 0 <= k <= n.
Conjecture 2: T(n,k) = T(n-k,0) + Sum_{i=1..n-k} T(n-k,i) * T(i+k,k) * k / (i + k - 1) for 0 < k <= n.

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0       1       2      3      4     5    6   7  8  9
================================================================
  0 :       1
  1 :       0       1
  2 :       1       1       1
  3 :       2       4       2      1
  4 :       9      15       9      3      1
  5 :      44      76      44     16      4     1
  6 :     265     455     265     95     25     5    1
  7 :    1854    3186    1854    666    174    36    6   1
  8 :   14833   25487   14833   5327   1393   287   49   7  1
  9 :  133496  229384  133496  47944  12536  2584  440  64  8  1
  etc.

Crossrefs

Cf. A000166 (column 0 and 2), A002467 (column 1), A006347 (column 3), A006348 (column 4), A009179 (row sums, signed), A352988 (matrix inverse).
Cf. A094587, A097807, A326326.

Programs

  • Maple
    T := proc(n,k) option remember;
if k > n then 0 else n * T(n-1,k) + (-1)^(n-k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 11 2022

Formula

T(n,n) = 1 for n >= 0.
T(n,n-1) = n - 1 for n > 0.
T(n,n-2) = (n - 1)^2 for n > 1.
T(n,0) = A000166(n) for n >= 0.
T(n,1) = A002467(n) for n > 0.
T(n,2) = A000166(n) for n > 1.
T(n,k) + T(n,k+1) = (n!) / (k!) for 0 <= k <= n.
T(n,k) = (n - 1) * (T(n-1,k) + T(n-2,k)) for 0 <= k < n-1.
T(n,k) = (T(n,k-2) - (k - 2) * T(n,k-1)) / (k - 1) for 1 < k <= n.
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (n - 1) * (p(n-1,x) + p(n-2,x)) + x^n for n > 0 with initial value p(0,x) = 1.
Row sums are p(n,1) = abs(A009179(n)) for n >= 0.
Alternating row sums are p(n,-1) = (-1)^n for n >= 0.
T(n,k) * T(n+1,k+1) - T(n+1,k) * T(n,k+1) = (-1)^(n-k) * A094587(n,k) for 0 <= k <= n.
Define 3x3-matrices T(i,j) with n <= i <= n+2 and k <= j <= k+2. Then we have: det(T(i,j)) = 0^(n-k) for 0 <= k <= n.
E.g.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n / (n!) = (Sum_{n>=k} (-t)^n / (n!)) * (-1)^k / (1 - t).
E.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n / (n!) = (x * exp(x * t) + exp(-t)) / ((1 + x) * (1 - t)).
p(n,x) = Sum_{k=0..n} ((n!)/(k!))*(x^(k+1) + (-1)^k)/(x + 1) for n >= 0.
T(n,k) = Sum_{i=0..n-k} (-1)^i * (n!) / ((k+i)!) for 0 <= k <= n.
T(n,k) equals matrix product of A094587 and A097807.

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).

A128181 A007318 * A128179 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 8, 9, 4, 8, 16, 21, 16, 5, 16, 32, 45, 44, 25, 6, 32, 64, 93, 104, 80, 36, 7, 64, 128, 189, 228, 210, 132, 49, 8, 128, 256, 381, 480, 495, 384, 203, 64, 9, 256, 512, 765, 988, 1095, 978, 651, 296, 81, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Binomial transform of A128179.
Row sums = A058396 starting (1, 3, 9, 25, 66, 168, ...).

Examples

			First few rows of the triangle:
   1;
   1,  2;
   2,  4,  3;
   4,  8,  9,   4;
   8, 16, 21,  16,  5;
  16, 32, 45,  44, 25,  6;
  32, 64, 93, 104, 80, 36, 7;
  ...

Crossrefs

Cf. A128179, A097807, A002260, A058396.
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