A008292 -id:A008292 - cp's OEIS frontend

A176488 Triangle T(n,k) = A008292(n+1,k+1) + A176487(n,k) - 1, 0<=k<=n.

1, 1, 1, 1, 8, 1, 1, 23, 23, 1, 1, 54, 136, 54, 1, 1, 117, 612, 612, 117, 1, 1, 244, 2395, 4850, 2395, 244, 1, 1, 499, 8605, 31271, 31271, 8605, 499, 1, 1, 1010, 29242, 176522, 312448, 176522, 29242, 1010, 1, 1, 2033, 95714, 910466, 2620832, 2620832, 910466

Offset: 0

Roger L. Bagula, Apr 19 2010

Row sums are 1, 2, 10, 48, 246, 1460, 10130, 80752, 725998, 7258092, 79834602,....

Apparently the row sums obey (-45*n+124)*s(n) +(45*n^2+127*n-654)*s(n-1) +(-206*n^2+227*n+708)*s(n-2) +(303*n^2-869*n+458)*s(n-3) -2*(71*n-125)*(n-2)*s(n-4)=0. - R. J. Mathar, Jun 16 2015

			1;
1, 1;
1, 8, 1;
1, 23, 23, 1;
1, 54, 136, 54, 1;
1, 117, 612, 612, 117, 1;
1, 244, 2395, 4850, 2395, 244, 1;
1, 499, 8605, 31271, 31271, 8605, 499, 1;
1, 1010, 29242, 176522, 312448, 176522, 29242, 1010, 1;
1, 2033, 95714, 910466, 2620832, 2620832, 910466, 95714, 2033, 1;
1, 4080, 305317, 4407094, 19476436, 31448746, 19476436, 4407094, 305317, 4080, 1;

Cf. A007318, A008292.

A176488 := proc(n,k)
    A008292(n+1,k+1)+A176487(n,k)-1 ;
end proc: # R. J. Mathar, Jun 16 2015

<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 1] + t[n, m, q - 2] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A014449 Numbers in the triangle of Eulerian numbers (A008292) that are not 1.

Original entry on oeis.org

4, 11, 11, 26, 66, 26, 57, 302, 302, 57, 120, 1191, 2416, 1191, 120, 247, 4293, 15619, 15619, 4293, 247, 502, 14608, 88234, 156190, 88234, 14608, 502, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 2036, 152637, 2203488, 9738114

Offset: 1

Mohammad K. Azarian

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

More terms from James Sellers

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A038675 Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).

Original entry on oeis.org

1, 1, 3, 1, 16, 10, 1, 55, 165, 35, 1, 156, 1386, 1456, 126, 1, 399, 8456, 25368, 11970, 462, 1, 960, 42876, 289920, 393030, 95040, 1716, 1, 2223, 193185, 2577135, 7731405, 5525091, 741741, 6435, 1, 5020, 803440, 19411480, 111675850, 176644468

Offset: 1

Alford Arnold

Andrews, Theory of Partitions, (1976), discussion of multisets.

Let a = a_1,a_2,...,a_n be a sequence on the alphabet {1,2,...,n}. Scan a from left to right and create an n-permutation by noting the POSITION of the elements as you come to them in order from least to greatest. See example. T(n,k) is the number of sequences that correspond to such a permutation having exactly n-k descents. [From Geoffrey Critzer, May 19 2010]

			1;
1,3;
1,16,10;
1,55,165,35;
1,156,1386,1456,126;
...
If a = 3,1,1,2,4,3 the corresponding 6-permutation is 2,3,4,1,6,5 because the first 1 is in the 2nd position, the second 1 is in the 3rd position,the 2 is in the 4th position, the first 3 is in the first position, the next 3 is in the 6th position and the 4 is in the 5th position of the sequence a. [From _Geoffrey Critzer_, May 19 2010]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, Mass., 1994, p. 269 (Worpitzky's identity).
Miklos Bona, Combinatorics of Permutations,Chapman and Hall,2004,page 6. [From Geoffrey Critzer, May 19 2010]

Cf. A001700, A014449, A000312.
Row sums yield A000312 (Worpitzky's identity).
Cf. A008292.

A:=(n,k)->sum((-1)^j*(k-j)^n*binomial(n+1,j),j=0..k): T:=(n,k)->A(n,k)*binomial(n+k-1,n): seq(seq(T(n,k),k=1..n),n=1..10);

Table[Table[Eulerian[n, k] Binomial[n + k, n], {k, 0, n - 1}], {n, 1,10}] (* Geoffrey Critzer, Jun 13 2013 *)

More terms from Emeric Deutsch, May 08 2004

A085503 Sub-triangle of A008292: take every second term of every second row.

Original entry on oeis.org

1, 1, 1, 1, 66, 1, 1, 1191, 1191, 1, 1, 14608, 156190, 14608, 1, 1, 152637, 9738114, 9738114, 152637, 1, 1, 1479726, 423281535, 2275172004, 423281535, 1479726, 1, 1, 13824739, 15041229521, 311387598411

Offset: 0

Roger L. Bagula, Dec 13 2010

			{1},
{1, 1},
{1, 66, 1},
{1, 1191, 1191, 1},
{1, 14608, 156190, 14608, 1},
{1, 152637, 9738114, 9738114, 152637, 1},

Claire Levaillant, Powers of two weighted sum of the first p divided Bernoulli numbers modulo p, arXiv:2001.03471 [math.CO], 2020. See p. 6.

Cf. A008292.

<< DiscreteMath`Combinatorica`
a = Table[Table[Eulerian[n + 1, 2*m], {m, 0, Floor[n/2]}], {n, 0, 20, 2}];
Flatten[%]

A141689 Average of Eulerian numbers (A008292) and Pascal's triangle (A007318): t(n,m) = (A008292(n,m) + A007318(n,m))/2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 36, 15, 1, 1, 31, 156, 156, 31, 1, 1, 63, 603, 1218, 603, 63, 1, 1, 127, 2157, 7827, 7827, 2157, 127, 1, 1, 255, 7318, 44145, 78130, 44145, 7318, 255, 1, 1, 511, 23938, 227638, 655240, 655240, 227638, 23938, 511, 1

Offset: 1

Roger L. Bagula, Sep 09 2008

Row sums are: {1, 2, 5, 16, 68, 376, 2552, 20224, 181568, 1814656, ...}.

If Pascal's triangle and the Eulerian numbers are both fundamental arrays, then there should be a combinatorial set "between" them.

			{1},
{1, 1},
{1, 3, 1},
{1, 7, 7, 1},
{1, 15, 36, 15, 1},
{1, 31, 156, 156, 31, 1},
{1, 63, 603, 1218, 603, 63, 1},
{1, 127, 2157, 7827, 7827, 2157, 127, 1},
{1, 255, 7318, 44145, 78130, 44145, 7318, 255, 1},
{1, 511, 23938, 227638, 655240, 655240, 227638, 23938, 511, 1}

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A008292, A007318.

Table[Table[(Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}] + Binomial[n - 1, k])/2, {k, 0, n - 1}], {n, 1, 10}]; Flatten[%]

Edited by N. J. A. Sloane, Dec 13 2008

A141690 Triangle t(n,m) = 2*A008292(n+1,m+1) - A007318(n,m), a linear combination of Eulerian numbers and Pascal's triangle, 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 126, 48, 1, 1, 109, 594, 594, 109, 1, 1, 234, 2367, 4812, 2367, 234, 1, 1, 487, 8565, 31203, 31203, 8565, 487, 1, 1, 996, 29188, 176412, 312310, 176412, 29188, 996, 1, 1, 2017, 95644, 910300, 2620582, 2620582, 910300

Offset: 0

Roger L. Bagula, Sep 09 2008

Row sums are 1, 2, 8, 40, 224, 1408, 10016, 80512, 725504, 7257088, ... = 2*(n+1)! - 2^n.

			Triangle begins
  1;
  1,    1;
  1,    6,     1;
  1,   19,    19,      1;
  1,   48,   126,     48,       1;
  1,  109,   594,    594,     109,       1;
  1,  234,  2367,   4812,    2367,     234,      1;
  1,  487,  8565,  31203,   31203,    8565,    487,     1;
  1,  996, 29188, 176412,  312310,  176412,  29188,   996,    1;
  1, 2017, 95644, 910300, 2620582, 2620582, 910300, 95644, 2017, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A008292, A007318.

A141690 := proc(n,m)
        2*A008292(n+1,m+1)-binomial(n,m) ;
end proc: # R. J. Mathar, Jul 12 2012

Table[Table[(2*Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}] - Binomial[n - 1, k]), {k, 0, n - 1}], {n, 1, 10}]; Flatten[%]

A143505 Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 11, 14, 23, 14, 11, 1, 1, 26, 70, 104, 139, 104, 70, 26, 1, 1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1, 1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1, 1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Row sums yield A000670 (without leading 1).

			Triangle begins:
   1;
   1,  1,   1;
   1,  4,   3,   4,   1;
   1, 11,  14,  23,  14,  11,   1;
   1, 26,  70, 104, 139, 104,  70,  26,   1;
   1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

Eric Weisstein's World of Mathematics, Polylogarithm

Compare with A141720.
Cf. A008292.
Cf. A143506, A143507.

Table[CoefficientList[FullSimplify[ExpandAll[(1 - x - 1/x)^(n + 1)*x^(n - 1)*PolyLog[-n, x + 1/x]/(x + 1/x)]], x], {n, 1, 10}]//Flatten

Row n is generated by the polynomial (1 - x - 1/x)^(n + 1)*x^(n - 1)*Li(-n, x + 1/x)/(x + 1/x), where Li(n, z) is the polylogarithm function.

E.g.f.: (exp(x*y) - exp((1 + x^2)*y))/(x*exp((1 + x^2)*y) - (1 + x^2)*exp(x*y)). - Franck Maminirina Ramaharo, Oct 25 2018

Edited and new name by Franck Maminirina Ramaharo, Oct 25 2018

A157117 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(nk + 1, n-k) if k <= n otherwise A008292(n(n-k), k), read by rows.

Original entry on oeis.org

2, 1, 1, 1, 8, 1, 1, 131, 131, 1, 1, 8204, 29216, 8204, 1, 1, 2097187, 44136233, 44136233, 2097187, 1, 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1, 1, 8796093022411, 150092195453359483, 288159195861579519, 288159195861579519, 150092195453359483, 8796093022411, 1

Offset: 0

Roger L. Bagula, Feb 23 2009

			  2;
  1,          1;
  1,          8,            1;
  1,        131,          131,            1;
  1,       8204,        29216,         8204,            1;
  1,    2097187,     44136233,     44136233,      2097187,          1;
  1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A008292, A157114, A157118.

Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
f:= func< n,k | k le n select Eulerian(n*k+1,n-k) else Eulerian(n*(n-k)+1, k) >;
A157117:= func< n,k | f(n,k) + f(n,n-k) >;
[A157117(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022

f[n_, k_]:= If[k<=n, Eulerian[n*k+1,n-k], Eulerian[n*(n-k)+1,k]];
T[n_, k_]:= f[n,k] + f[n,n-k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def f(n,k): return Eulerian(n*k+1,n-k) if (kA157117(n,k): return f(n,k) + f(n,n-k)
flatten([[A157117(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k).

T(n, n-k) = T(n, k).

Edited by G. C. Greubel, Jan 11 2022

A159346 Triangle read by rows: every third term of every third row of A008292.

Original entry on oeis.org

1, 1, 1, 1, 2416, 1, 1, 455192, 455192, 1, 1, 45533450, 2275172004, 45533450, 1, 1, 3572085255, 3207483178157, 3207483178157, 3572085255, 1, 1, 251732291184, 2527925001876036, 37307713155613000, 2527925001876036, 251732291184, 1, 1, 16871482830550, 1454842842001939656

Offset: 0

Roger L. Bagula, Dec 13 2010

			Triangle begins:
  {1},
  {1, 1},
  {1, 2416, 1},
  {1, 455192, 455192, 1},
  {1, 45533450, 2275172004, 45533450, 1},
  {1, 3572085255, 3207483178157, 3207483178157, 3572085255, 1},
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A008292, A085503.

<< DiscreteMath`Combinatorica`
k = 3;
a = Table[Table[Eulerian[n + 1, k*m], {m, 0, Floor[n/k]}], {n, 0, 10*k,k}];
Flatten[%]

T(n, k) = A008292(3*n+1, 3*k+1). - Jason Yuen, Feb 04 2025

Edited by N. J. A. Sloane, Jan 01 2011

A014469 Triangular array formed from odd elements to right of middle of rows of the triangle of Eulerian numbers (A008292).

Original entry on oeis.org

1, 1, 11, 1, 1, 57, 1, 1191, 1, 15619, 4293, 247, 1, 1, 1013, 1, 152637, 1, 10187685, 478271, 4083, 1, 423281535, 1, 12843262863, 2571742175, 16369, 1, 311387598411, 15041229521, 13824739, 1, 6382798925475, 3207483178157, 782115518299

Offset: 1

Mohammad K. Azarian

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

More terms from James Sellers

cp's OEIS Frontend

A176488 Triangle T(n,k) = A008292(n+1,k+1) + A176487(n,k) - 1, 0<=k<=n.

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A014449 Numbers in the triangle of Eulerian numbers (A008292) that are not 1.

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A038675 Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).

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A085503 Sub-triangle of A008292: take every second term of every second row.

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A141689 Average of Eulerian numbers (A008292) and Pascal's triangle (A007318): t(n,m) = (A008292(n,m) + A007318(n,m))/2.

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A141690 Triangle t(n,m) = 2*A008292(n+1,m+1) - A007318(n,m), a linear combination of Eulerian numbers and Pascal's triangle, 0 <= m <= n.

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Maple

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A143505 Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

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A157117 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k), read by rows.

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A157117 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(nk + 1, n-k) if k <= n otherwise A008292(n(n-k), k), read by rows.