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

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

Offset: 8

Author

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

Keywords

• nonn
• easy

Comments

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

References

• 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).

Links

• 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).

Crossrefs

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).

Programs

Magma

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

Mathematica

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 *)

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))

Python

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

Formula

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

Extensions

More terms from Christian G. Bower, May 12 2000