A008292 -id:A008292 - cp's OEIS frontend

A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 71, 29, 1, 1, 61, 311, 311, 61, 1, 1, 125, 1205, 2435, 1205, 125, 1, 1, 253, 4313, 15653, 15653, 4313, 253, 1, 1, 509, 14635, 88289, 156259, 88289, 14635, 509, 1, 1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021, 1

Offset: 0

Roger L. Bagula, Apr 19 2010

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   13,    13,      1;
  1,   29,    71,     29,       1;
  1,   61,   311,    311,      61,       1;
  1,  125,  1205,   2435,    1205,     125,      1;
  1,  253,  4313,  15653,   15653,    4313,    253,     1;
  1,  509, 14635,  88289,  156259,   88289,  14635,   509,    1;
  1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000182, A007318, A008292, A033312, A036563, A141689.

A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;
[A176487(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 31 2024

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

Needs["Combinatorica`"];
T[n_, k_, 0]:= Binomial[n, k];
T[n_, k_, 1]:= Eulerian[1 + n, k];
T[n_, k_, q_]:= T[n,k,q] = T[n,k,q-1] + T[n,k,q-2] - 1;
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A176487(n,k): return binomial(n,k) +eulerian_number(n+1,k) -1
print(flatten([[A176487(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 31 2024

T(n, k) = A007318(n,k) + A008292(n+1,k+1) - 1, 0 <= k <= n.
Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).
T(n, k) = 2*A141689(n+1,k+1) - 1. - R. J. Mathar, Jan 19 2011
From G. C. Greubel, Dec 31 2024: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A036563(n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = ((-1)^(n/2)*A000182(n/2 + 1) - 1)*(1 + (-1)^n)/2 + [n=0]. (End)

A001244 Eulerian numbers (Euler's triangle: column k=8 of A008292, column k=7 of A173018).

Original entry on oeis.org

1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472, 6382798925475, 83137223185370, 1006709967915228, 11485644635009424, 124748182104463860, 1300365805079109480, 13093713503185076040

Offset: 8

N. J. A. Sloane, Mira Bernstein, and Robert G. Wilson v

There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)
For the general computation of the o.g.f. and e.g.f. see A123125. - Wolfdieter Lang, Apr 03 2017

L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." §6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 2601.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 8..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
Index entries for linear recurrences with constant coefficients, signature (120, -6930, 256564, -6843837, 140161164, -2293167668, 30793317984, -346027498674, 3301174490432, -27034426023228, 191677191769368, -1184495927428914, 6413285791562760, -30547549870770240, 128399094121475760, -477325107218885805, 1571764443755152680, -4588173158058601250, 11875425392771515860, -27240699344951953809, 55318442559624109580, -99273350219483495580, 157041371328829338576, -218253110396224153888, 265336916554318663296, -280638192440433919872, 256449901319079809536, -200704456428999204096, 133025721255740648448, -73584771640934648832, 33313567375875428352, -12012672014150270976, 3315383509586411520, -657169361790566400, 83234996748288000, -5056584744960000).

Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)
Cf. A123125 (row reversed version of A173018).
Cf. A000012, A000460, A000498, A000505, A000514, A001243 (columns for smaller k).

A001244:= func< n | EulerianNumber(n,7) >;
[A001244(n): n in [8..40]]; // G. C. Greubel, Dec 31 2024

k = 8; Table[k^(n + k - 1) + Sum[(-1)^i/i!*(k - i)^(n + k - 1) * Product[n + k + 1 - j, {j, 1, i}], {i, 1, k - 1}], {n, 1, 15}] (* Michael De Vlieger, Aug 04 2015, after PARI *)

A001244(n)=8^(n+8-1)+sum(i=1,8-1,(-1)^i/i!*(8-i)^(n+8-1)*prod(j=1,i,n+8+1-j))

from sage.combinat.combinat import eulerian_number
print([eulerian_number(n,7) for n in range(8,41)]) # G. C. Greubel, Dec 31 2024

a(n) = 8^(n+8-1) + Sum_{i=1..8-1} ((-1)^i/i!)*(8-i)^(n+8-1) * Product_{j=1..i} (n+8+1 - j). - Randall L Rathbun, Jan 23 2002

a(n) = k^n + Sum_{j=1..k-1} (-1)^j*binomial(n+1,j)*(k-j)^n, with k = 8, for n >= 8. - G. C. Greubel, Dec 31 2024

More terms from Christian G. Bower, May 12 2000

A154694 Triangle read by rows: T(n,k) = ((3/2)^k2^n + (2/3)^k3^n)*A008292(n+1,k+1).

Original entry on oeis.org

2, 5, 5, 13, 48, 13, 35, 330, 330, 35, 97, 2028, 4752, 2028, 97, 275, 11970, 54360, 54360, 11970, 275, 793, 69840, 557388, 1043712, 557388, 69840, 793, 2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315, 6817, 2388516, 51011136, 247761072, 404844480, 247761072, 51011136, 2388516, 6817

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

			Triangle begins as:
     2;
     5,      5;
    13,     48,      13;
    35,    330,     330,       35;
    97,   2028,    4752,     2028,       97;
   275,  11970,   54360,    54360,    11970,     275;
   793,  69840,  557388,  1043712,   557388,   69840,    793;
  2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A 34 (3) (1986) 1986, page 2502, (FIG. 3)

Cf. A004123 (row sums), A154693, A256890.

A154694:= func< n,k | (2^(n-k)*3^k+2^k*3^(n-k))*EulerianNumber(n+1, k) >;
[A154694(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154694 := proc(n,m)
    (3^m*2^(n-m)+2^m*3^(n-m))*A008292(n+1,m+1) ;
end proc:
seq(seq( A154694(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Mar 11 2024

T[n_, k_, p_, q_] := (p^(n - k)*q^k + p^k*q^(n - k))*Eulerian[n+1,k];
Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten

from sage.all import *
from sage.combinat.combinat import eulerian_number
def A154694(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*eulerian_number(n+1,k)
print(flatten([[A154694(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Sum_{k=0..n} T(n, k) = A004123(n+2).

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

A055325 Matrix inverse of Euler's triangle A008292.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -23, 33, -11, 1, 425, -620, 220, -26, 1, -18129, 26525, -9520, 1180, -57, 1, 1721419, -2519664, 905765, -113050, 5649, -120, 1, -353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1, 153923102577

Offset: 1

Christian G. Bower, May 12 2000

			Triangle starts:
  [1]         1;
  [2]        -1,         1;
  [3]         3,        -4,          1;
  [4]       -23,        33,        -11,        1;
  [5]       425,      -620,        220,      -26,        1;
  [6]    -18129,     26525,      -9520,     1180,      -57,     1;
  [7]   1721419,  -2519664,     905765,  -113050,     5649,  -120,    1;
  [8]-353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1;

Robert Israel, Table of n, a(n) for n = 1..3321 (rows 1 to 81, flattened)

Cf. A008292, A162498.

A008292:= proc(n, k) option remember;
  if k < 1 or k > n then 0
  elif k = 1 or k = n then 1
  else (k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1))
  fi
end proc:
T:= Matrix(10,10,(i,j) -> A008292(i,j)):
R:= T^(-1):
seq(seq(R[i,j],j=1..i),i=1..10); # Robert Israel, May 25 2018

m = 10 (*rows*);
t[n_, k_] := Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
M = Array[t, {m, m}] // Inverse;
Table[M[[i, j]], {i, 1, m}, {j, 1, i}] // Flatten (* Jean-François Alcover, Mar 05 2019 *)
T[1, 1] := 1; T[n_, k_]/;1<=k<=n := T[n, k] = (n-k+1) T[n-1, k-1] + k T[n-1, k]; T[n_, k_] := 0;(*A008292*)
iT[n_, n_]/;n>=1 := 1; iT[n_, k_]/;1<=kA055325*)
Flatten@Table[iT[n, k], {n, 1, 9}, {k, 1, n}] (* Oliver Seipel, Feb 10 2025 *)

A141686 Triangle read by rows: T(n, k) = binomial(n-1, k-1)*A008292(n, k).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 33, 33, 1, 1, 104, 396, 104, 1, 1, 285, 3020, 3020, 285, 1, 1, 720, 17865, 48320, 17865, 720, 1, 1, 1729, 90153, 546665, 546665, 90153, 1729, 1, 1, 4016, 409024, 4941104, 10933300, 4941104, 409024, 4016, 1, 1, 9117, 1722240, 38236128, 165104604, 165104604, 38236128, 1722240, 9117, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 08 2008

			Triangle begins as:
  1;
  1,    1;
  1,    8,       1;
  1,   33,      33,        1;
  1,  104,     396,      104,         1;
  1,  285,    3020,     3020,       285,         1;
  1,  720,   17865,    48320,     17865,       720,        1;
  1, 1729,   90153,   546665,    546665,     90153,     1729,       1;
  1, 4016,  409024,  4941104,  10933300,   4941104,   409024,    4016,    1;
  1, 9117, 1722240, 38236128, 165104604, 165104604, 38236128, 1722240, 9117, 1;

Reinhard Zumkeller, Rows n = 1..125 of table, flattened

Cf. A007318, A008292, A104098.

a141686 n k = a141686_tabl !! (n-1) !! (k-1)
a141686_row n = a141686_tabl !! (n-1)
a141686_tabl = zipWith (zipWith (*)) a007318_tabl a008292_tabl
-- Reinhard Zumkeller, Apr 16 2014

A141686:= func< n, k | Binomial(n-1, k-1)*EulerianNumber(n, k-1) >;
[A141686(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 31 2024

(* Recurrence for A008292 *)
f[n_, k_]:= If[k==1||k==n,1, (n-k+1)*f[n-1,k-1] + k*f[n-1,k]];
Table[f[n, k]*Binomial[n-1,k-1], {n,12}, {k,n}]//Flatten
(* Second program *)
Needs["Combinatorica`"];
A141686[n_, k_]:= Binomial[n-1,k-1]*Eulerian[n,k-1];
Table[A141686[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Dec 31 2024 *)

# or SageMath
from sage.combinat.combinat import eulerian_number
def A141686(n,k): return binomial(n-1,k-1)*eulerian_number(n,k-1)
print(flatten([[A141686(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Dec 31 2024

T(n, k) = T(n, n-k+1).

Sum_{k=1..n} T(n, k) = A104098(n) (row sums).

keyword:tabl inserted, indices corrected by the Assoc. Eds. of the OEIS, Jun 30 2010

A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.

Original entry on oeis.org

1, 1, 2, 18, 244, 5710, 188908, 8702820, 524888040, 40393084950, 3853034107900, 446671026849916, 61824801560228056, 10072685383683311116, 1907978676359896992824, 415795605119514578204616, 103294156408291202467520976, 29018125910193347265466916070

Offset: 0

Paul D. Hanna, Nov 29 2009

nonn

Row sums of Eulerian triangle A008292 yield the factorials.

			a(1) = 1 = 1;
a(2) = 1 + 1 = 2;
a(3) = 1 + 4^2 + 1 = 18;
a(4) = 1 + 11^2 + 11^2 + 1 = 244;
a(5) = 1 + 26^2 + 66^2 + 26^2 + 1 = 5710;
a(6) = 1 + 57^2 + 302^2 + 302^2 + 57^2 + 1 = 188908.

Alois P. Heinz, Table of n, a(n) for n = 0..253

Cf. A008292.

Column k=2 of A335545.

a:= n-> add(combinat[eulerian1](n, k)^2, k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Sep 10 2020

{a(n)=sum(k=0,n,sum(j=0, k, (-1)^j*(k-j)^n*binomial(n+1, j))^2)}

a(n) = Sum_{k=0..n} [ Sum_{j=0..k} (-1)^j*(k-j)^n*C(n+1, j) ]^2.

A176490 Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 33, 33, 1, 1, 101, 295, 101, 1, 1, 293, 1983, 1983, 293, 1, 1, 841, 11733, 25963, 11733, 841, 1, 1, 2425, 64949, 275341, 275341, 64949, 2425, 1, 1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1, 1, 20685, 1804179, 22163163

Offset: 0

Roger L. Bagula, Apr 19 2010

Row sums are: 1, 2, 11, 68, 499, 4554, 51113, 685432, 10684791, 189423350, 3755807989,....

Conjecture on the row sums s(n): 859*(n+1)*s(n) +(-2577*n^2-15955*n+33324)*s(n-1) +(1718*n^3+39275*n^2-102106*n-16383)*s(n-2) +(-25038*n^3+35127*n^2+252701*n-453082)*s(n-3) +(n-3)*(57834*n^2-211893*n+212386)*s(n-4) -2*(17257*n-29530)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 16 2015

			1;
1, 1;
1, 9, 1;
1, 33, 33, 1;
1, 101, 295, 101, 1;
1, 293, 1983, 1983, 293, 1;
1, 841, 11733, 25963, 11733, 841, 1;
1, 2425, 64949, 275341, 275341, 64949, 2425, 1;
1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1;
1, 20685, 1804179, 22163163, 70723647, 70723647, 22163163, 1804179, 20685, 1;
1, 61073, 9268777, 180504391, 916661395, 1542816715, 916661395, 180504391, 9268777, 61073, 1;

Cf. A007318, A008292, A060187, A176487.

A176490 := proc(n,k)
    A008292(n+1,k+1)+A060187(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015

(*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A065826 Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.

Original entry on oeis.org

1, 1, 2, 1, 8, 3, 1, 22, 33, 4, 1, 52, 198, 104, 5, 1, 114, 906, 1208, 285, 6, 1, 240, 3573, 9664, 5955, 720, 7, 1, 494, 12879, 62476, 78095, 25758, 1729, 8, 1, 1004, 43824, 352936, 780950, 529404, 102256, 4016, 9, 1, 2026, 143520, 1820768, 6551770, 7862124, 3186344, 382720, 9117, 10

Offset: 1

Henry Bottomley, Dec 06 2001

Row sums are (n+1)!/2, i.e., A001710 offset, implying that if n balls are put at random into n boxes, the expected number of boxes with at least one ball is (n+1)/2 and the expected number of empty boxes is (n-1)/2.

T(n,k) is the number of permutations of {1,2,...,n+1} that start with an ascent and that have k-1 descents. - Ira M. Gessel, May 02 2017

			Rows start:
  1;
  1,   2;
  1,   8,     3;
  1,  22,    33,     4;
  1,  52,   198,   104,     5;
  1, 114,   906,  1208,   285,     6;
  1, 240,  3573,  9664,  5955,   720,    7;
  1, 494, 12879, 62476, 78095, 25758, 1729, 8;
  etc.

Alois P. Heinz, Rows n = 1..141, flattened
Ron M. Adin, Sergi Elizalde, Victor Reiner, Yuval Roichman, Cyclic descent extensions and distributions, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.
Ron M. Adin, Victor Reiner, Yuval Roichman, On cyclic descents for tableaux, arXiv:1710.06664 [math.CO], 2017.

Cf. A008292.

Flat(List([1..10],n->List([1..n],k->Sum([0..k],j->k*(-1)^j*(k-j)^n*Binomial(n+1,j))))); # Muniru A Asiru, Mar 09 2019

T:=(n,k)->add(k*(-1)^j*(k-j)^n*binomial(n+1,j),j=0..k): seq(seq(T(n,k),k=1..n),n=1..10); # Muniru A Asiru, Mar 09 2019

Array[Range[Length@ #] # &@ CoefficientList[(1 - x)^(# + 1)*PolyLog[-#, x]/x, x] &, 10] (* Michael De Vlieger, Sep 24 2018, after Vaclav Kotesovec at A008292 *)

T(n, k) = k*(a(n-1, k) + a(n-1, k-1)*(n-k+1)/(k-1)) [with T(n, 1) = 1] = Sum_{j=0..k} k*(-1)^j*(k-j)^n*binomial(n+1, j).

E.g.f.: (exp(x*(1-t)) - 1 - x*(1-t))/((1-t)*(1 - t*exp(x*(1-t)))). - Ira M. Gessel, May 02 2017

A087674 Value of the n-th Eulerian polynomial (cf. A008292) evaluated at x=-2.

Original entry on oeis.org

1, 1, -1, -3, 15, 21, -441, 477, 19935, -101979, -1150281, 14838957, 60479055, -2328851979, 3529587879, 403992301437, -3333935426625, -72778393505979, 1413503392326039, 10851976875907917, -554279405351601105, 713848745428080021

Offset: 0

Vladeta Jovovic, Sep 26 2003

sign

			G.f. = 1 + x - x^2 - 3*x^3 + 15*x^4 + 21*x^5 - 441*x^6 + 477*x^7 + 19935*x^8 + ... - _Michael Somos_, Aug 27 2018

Cf. A000670, A212846.

Table[-3^(n+1)/2*PolyLog[-n, -2], {n, 0, 21}] (* Jean-François Alcover, Apr 26 2013 *)
a[ n_] := If[ n < 0, 0, n! 3/2 SeriesCoefficient[ 1 / (1 + Exp[-3 x] / 2), {x, 0, n}]]; (* Michael Somos, Aug 27 2018 *)

{a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1+3*k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

x='x+O('x^66); Vec(serlaplace( 3*exp(x)/(2*exp(x)+exp(-2*x)) )) \\ Joerg Arndt, Apr 21 2013

{a(n) = if( n<0, 0, n! * 3/2 * polcoeff( 1 / (1 + exp( -3*x + x * O(x^n)) / 2), n))}; /* Michael Somos, Aug 27 2018 */

a(n) = -3^(n+1)/2*polylog(-n, -2).
E.g.f.: 3*exp(x)/(2*exp(x)+exp(-2*x)).
O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 3*k*x). [Paul D. Hanna, Jul 20 2011]
G.f.: 1/Q(0), where Q(k)= 1 - x*(k+1)/(1 + x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
a(n) = (-1)^n * A212846(n). - Michael Somos, Aug 27 2018
a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * (-3)^(n-k). - Ilya Gutkovskiy, Jun 08 2022

More terms from Paul D. Hanna, Jul 20 2011

A104098 a(n) = Sum_{k=1..n} binomial(n-1, k-1)*A008292(n, k) for n >= 1.

Original entry on oeis.org

1, 2, 10, 68, 606, 6612, 85492, 1277096, 21641590, 410144180, 8595133548, 197346180792, 4926442358124, 132847425483528, 3848398710032616, 119187270233781456, 3929892162743796390, 137444081992905303540, 5082053733073190713660, 198081684441819323760920

Offset: 1

Paul D. Hanna, Mar 29 2006

nonn

			   1 = 1*1
   2 = 1*1 +  1*1
  10 = 1*1 +  2*4 + 1*1
  68 = 1*1 + 3*11 + 3*11 + 1*1
  ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..201

Cf. A008292, A011818.

a := n -> local k; add(binomial(n - 1, k - 1) * combinat:-eulerian1(n, k - 1), k = 1..n): seq(a(n), n = 1..20);  # Peter Luschny, Oct 29 2023

Table[Sum[Binomial[n-1,k-1] * Sum[(-1)^j * (k-j)^n * Binomial[n+1,j], {j, 0, k}], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 09 2020 *)

a(n) ~ sqrt(3) * 2^(n-1) * n^n / exp(n). - Vaclav Kotesovec, Oct 09 2020, modified for offset 1, Oct 29 2023

Offset changed by Georg Fischer, Oct 29 2023

cp's OEIS Frontend

A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

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A001244 Eulerian numbers (Euler's triangle: column k=8 of A008292, column k=7 of A173018).

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A154694 Triangle read by rows: T(n,k) = ((3/2)^k*2^n + (2/3)^k*3^n)*A008292(n+1,k+1).

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A055325 Matrix inverse of Euler's triangle A008292.

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A141686 Triangle read by rows: T(n, k) = binomial(n-1, k-1)*A008292(n, k).

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A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.

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A176490 Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

A154694 Triangle read by rows: T(n,k) = ((3/2)^k2^n + (2/3)^k3^n)*A008292(n+1,k+1).