A047682 -id:A047682 - cp's OEIS frontend

A211233 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.

1, 2, 3, 1, 4, 10, 4, 1, 1, 7, 27, 13, -13, -27, -7, -1, 1, 12, 69, 16, -182, -376, -182, 16, 69, 12, 1, 1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, -176, -21, -1, 1, 38, 456, -886, -8292, -20322, -6536, 35890, 65862, 35890, -6536, -20322, -8292, -886, 456, 38, 1

Offset: 1

N. J. A. Sloane, Apr 05 2012

			Triangle begins
  1,  2,   3;
  1,  4,  10,   4,     1;
  1,  7,  27,  13,   -13,   -27,    -7,   -1;
  1, 12,  69,  16,  -182,  -376,  -182,   16,   69,   12,  1;
  1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, ... ;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1366 (rows 1..30)
D. H. Lehmer, Generalized Eulerian numbers, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).

Row sums of even rows are A047682; row sums of odd rows are zero for n > 1.

Cf. A008292, A211232, A211234, A211235.

T(n,r=3)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}
{my(A=T(5)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, May 18 2020

From Andrew Howroyd, May 18 2020: (Start)
T(n,k) = k*T(n-1,k) - (n-k)*T(n-1,k-1) - (2*n-k)*T(n-1,k-2) - (3*n-k)*T(n-1,k-3) for n > 1.
A047682(n) = Sum_{k>=1} T(2*n, k).
(End)

Terms a(39) and beyond from Andrew Howroyd, May 18 2020

A241066 Array t(n,k) = k^(2n)(k^(2n)-1)BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 6, 16, 54, 20, 272, 2106, 544, 50, 7936, 179334, 66560, 3250, 105, 353792, 26414586, 17895424, 968750, 13986, 196, 22368256, 5957217414, 8329625600, 635781250, 8637840, 48020, 336, 1903757312, 1906398972666, 5937093935104, 722480468750, 11754617616, 54925276, 139776, 540

Offset: 1

Jean-François Alcover, Apr 16 2014

For any integers n and k, the ratio k^(2n)*(k^(2n)-1)*B(2n)/(2n) is always an integer.
Row 1 is A002415 = 4-D pyramidal numbers,
Row 2 and following rows are not in the OEIS,
Column 1 is A000182 = Tangent numbers,
Column 2 is A047681,
Column 3 is A047682,
Column 4 is A047683,
Column 5 and following columns are not in the OEIS.

			Array begins:
   1,        6,         20,           50,            105, ...
   2,       54,        544,         3250,          13986, ...
  16,     2106,      66560,       968750,        8637840, ...
272,   179334,   17895424,    635781250,    11754617616, ...
7936, 26414586, 8329625600, 722480468750, 27698169542400, ...
etc.

MathWorld, Bernoulli Number
Wikipedia, Bernoulli number

Cf. A000182, A002415, A047681, A047682, A047683.

nmax = 8; t[n_, k_] := k^(2*n)*(k^(2*n)-1)*BernoulliB[2*n]/(2*n); Table[t[n-k+2, k] // Abs, {n, 1, nmax}, {k, 2, n+1}] // Flatten

cp's OEIS Frontend

A211233 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.

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A241066 Array t(n,k) = k^(2n)(k^(2n)-1)BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.

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cp's OEIS Frontend

A211233 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A241066 Array t(n,k) = k^(2n)*(k^(2n)-1)*BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A241066 Array t(n,k) = k^(2n)(k^(2n)-1)BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.