A046802 -id:A046802 - cp's OEIS frontend

A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).

1, 2, 2, 3, 9, 3, 4, 28, 28, 4, 5, 75, 165, 75, 5, 6, 186, 786, 786, 186, 6, 7, 441, 3311, 6181, 3311, 441, 7, 8, 1016, 12888, 40888, 40888, 12888, 1016, 8, 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9, 10, 5110, 168670, 1312750, 3445510, 3445510, 1312750, 168670, 5110, 10

Offset: 0

Peter Luschny, Mar 11 2025

Consider A381706, the number of permutations of k chosen numbers in [n] with i-1 descents, as a sequence of squares of size 1x1, 2x2, 3x3, ..., as displayed in the example section of A381706. Conjecture: T(n, k) is the sum of column k+1 of the (n+1)th square; in other words: T(n, k) = Sum_{j=0..n} b(n+1, j+1, k+1).

			Triangle starts:
  [0] 1;
  [1] 2,    2;
  [2] 3,    9,     3;
  [3] 4,   28,    28,      4;
  [4] 5,   75,   165,     75,      5;
  [5] 6,  186,   786,    786,    186,      6;
  [6] 7,  441,  3311,   6181,   3311,    441,     7;
  [7] 8, 1016, 12888,  40888,  40888,  12888,  1016,    8;
  [8] 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9;

Cf. A046802, A173018 (Eulerian1), A122045 (Euler), A058877 (column 1), A007526 (row sums), A381706 (generalized Eulerian).

T := (n, k) -> (n + 1)*add(binomial(n, j)*combinat:-eulerian1(j, j - k), j = k .. n):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;
# Using the e.g.f.:
egf := ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1))))/(exp(x*(y - 1)) - y)^2:
ser := simplify(series(egf, x, 10)):
seq(seq(n!*coeff(coeff(ser, x, n), y, k), k = 0..n), n = 0..9);

# Using function eulerian1 from A173018.
def T(n: int, k: int) -> int:
    return (n + 1) * sum(binomial(n, j) * eulerian1(j, j-k) for j in (k..n))
def Trow(n) -> list[int]: return [T(n, k) for k in (0..n)]
for n in (0..8): print(f"{n}: ", Trow(n))

T(n, k) = n! * [y^k] [x^n] ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1)))) / (exp(x*(y - 1)) - y)^2.
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * (n + 1) * Euler(n).
T(n, k) = (n + 1) * A046802(n, k).

A168423 Triangle read by rows: expansion of (1 - x)/(exp(t)(1 - xexp(t*(1 - x)))).

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 7, 1, -1, 1, 1, 21, 21, 1, 1, -1, 1, 51, 161, 51, 1, -1, 1, 1, 113, 813, 813, 113, 1, 1, -1, 1, 239, 3361, 7631, 3361, 239, 1, -1, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, -1, 1, 1003, 42865, 320107, 607009, 320107, 42865

Offset: 0

Roger L. Bagula, Nov 25 2009

This sequence was derived from the Eulerian number umbral calculus expansion and A046802 by taking the exp(t) term and inverting it.
What is interesting here is the '1,-1' terms that appear.
I had thought I would get "1,5,1" not "1,7,1" from this function.
An OEIS search came up with A046739 which has the same internal symmetric number structure.
Inverse binomial transform of Eulerian numbers A123125. [Paul Barry, May 10 2011]

			{1},
{-1, 1},
{1, -1, 1},
{-1, 1, 1, 1},
{1, -1, 1, 7, 1},
{-1, 1, 1, 21, 21, 1},
{1, -1, 1, 51, 161, 51, 1},
{-1, 1, 1, 113, 813, 813, 113, 1},
{1, -1, 1, 239, 3361, 7631, 3361, 239, 1},
{-1, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1},
{1, -1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1}

Cf. A046802, A046739, A000166 (row sums), A123125.

p[t_] = (1 - x)/(Exp[t]*(1 - x*Exp[t*(1 - x)]))
a = Table[ CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];
Flatten[a]

E.g.f. sum(T(n,k) t^n/n! x^k) = p(x,t) = (1 - x)/(exp(t)*(1 - x*exp(t*(1 - x))))

T(n,k)=sum{j=0..n, (-1)^(n-j)*C(n,j)*A123125(j,k)}. [Paul Barry, May 10 2011]

A318143 Coefficients of the polynomials generated by the e.g.f. cosh(xz)(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 4, 4, 0, 1, 11, 17, 7, 1, 1, 26, 76, 66, 16, 0, 1, 57, 317, 467, 237, 31, 1, 1, 120, 1212, 2962, 2612, 806, 64, 0, 1, 247, 4321, 17215, 24145, 13519, 2641, 127, 1, 1, 502, 14644, 92554, 199192, 178486, 65884, 8434, 256, 0

Offset: 0

Peter Luschny, Aug 19 2018

			[n\k][0,   1,    2,     3,     4,     5,    6,   7,  8]
[0]   1;
[1]   1,   0;
[2]   1,   1,    1;
[3]   1,   4,    4,     0;
[4]   1,  11,   17,     7,     1;
[5]   1,  26,   76,    66,    16,     0;
[6]   1,  57,  317,   467,   237,    31,    1;
[7]   1, 120, 1212,  2962,  2612,   806,   64,   0;
[8]   1, 247, 4321, 17215, 24145, 13519, 2641, 127, 1;

Row sums are (-1)^n*A009179(n).
Alternating row sums are 1.
Polynomials evaluated at x = 0 are 1.
T(n, n-1) = A051049(n-1) for n >= 1.
T(n, 1) = A000295(n) for n >= 0.
Cf. A046802, A173018.

gf := cosh(x*z)*(x-1)/(x-exp(z*(x-1))):
ser := series(gf, z, 12): p := n -> normal(n!*coeff(ser, z, n)):
seq(seq(coeff(p(n),x,k), k=0..n), n=0..10);

A343804 T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 15, 11, 1, 1, 64, 96, 26, 1, 1, 325, 824, 448, 57, 1, 1, 1956, 7417, 6718, 1779, 120, 1, 1, 13699, 71595, 96633, 43411, 6429, 247, 1, 1, 109600, 746232, 1393588, 944618, 243928, 21898, 502, 1, 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

Offset: 0

Peter Luschny, Apr 30 2021

			Triangle starts:
[0] 1
[1] 1, 1
[2] 1, 4,      1
[3] 1, 15,     11,      1
[4] 1, 64,     96,      26,       1
[5] 1, 325,    824,     448,      57,       1
[6] 1, 1956,   7417,    6718,     1779,     120,     1
[7] 1, 13699,  71595,   96633,    43411,    6429,    247,     1
[8] 1, 109600, 746232,  1393588,  944618,   243928,  21898,   502,   1
[9] 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

Row sums: A084262.

Cf. A046802 (Eulerian first order).

T := (n, k) -> add(binomial(n, r)*combinat:-eulerian2(r, r-k), r = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9);

cp's OEIS Frontend

A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).

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A168423 Triangle read by rows: expansion of (1 - x)/(exp(t)(1 - xexp(t*(1 - x)))).

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A318143 Coefficients of the polynomials generated by the e.g.f. cosh(xz)(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.

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Examples

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Maple

A343804 T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Examples

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Programs

Maple

cp's OEIS Frontend

A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

SageMath

Formula

A168423 Triangle read by rows: expansion of (1 - x)/(exp(t)*(1 - x*exp(t*(1 - x)))).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A318143 Coefficients of the polynomials generated by the e.g.f. cosh(x*z)*(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A343804 T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A168423 Triangle read by rows: expansion of (1 - x)/(exp(t)(1 - xexp(t*(1 - x)))).

A318143 Coefficients of the polynomials generated by the e.g.f. cosh(xz)(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.