A008278 Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 15, 1, 1, 15, 65, 90, 31, 1, 1, 21, 140, 350, 301, 63, 1, 1, 28, 266, 1050, 1701, 966, 127, 1, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice

Comments

The n-th row also gives the coefficients of the sigma polynomial of the empty graph \bar K_n. - Eric W. Weisstein, Apr 07 2017

The n-th row also gives the coefficients of the independence polynomial of the (n-1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Apr 03 2018

From Gus Wiseman, Aug 11 2020: (Start)

Conjecture: also the number of divisors of the superprimorial A006939(n - 1) that have 0 <= k <= n distinct prime factors, all appearing with distinct multiplicities. For example, row n = 4 counts the following divisors of 360:

1 2 12 360

3 18

4 20

5 24

8 40

9 45

72

Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.

Crossrefs:

A006939 lists superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A076954 can be used instead of A006939.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A336420 is the version counting all prime factors, not just distinct ones.

Cf. A000005, A027423, A095149, A124010, A317829, A327498, A336419, A336421.

(End)

From Leonidas Liponis, Aug 26 2024: (Start)

It appears that this sequence is related to the combinatorial form of Faà di Bruno's formula. Specifically, the number of terms for the n-th derivative of a composite function y = f(g(x)) matches the number of partitions of n.

For example, consider the case where g(x) = e^x, in which all derivatives of g(x) are equal. The first 5 rows of A008278 appear as the factors of derivatives of f(x), highlighted here in brackets:

dy/dx = [ 1 ] * f'(e^x) * e^x

d^2y/dx^2 = [ 1 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^3y/dx^3 = [ 1 ] * f'''(e^x) * e^{3x} + [ 3 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^4y/dx^4 = [ 1 ] * f''''(e^x) * e^{4x} + [ 6 ] * f'''(e^x) * e^{3x} + [ 7 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^5y/dx^5 = [ 1 ] * f'''''(e^x) * e^{5x} + [ 10 ] * f''''(e^x) * e^{4x} + [ 25 ] * f'''(e^x) * e^{3x} + [ 15 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

This pattern is observed in Mathematica for the first 10 cases, using the code below.

(End)

Examples

			The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,    1;
  1, 10,  25,   15,    1;
  1, 15,  65,   90,   31,    1;
  1, 21, 140,  350,  301,   63,    1;
  1, 28, 266, 1050, 1701,  966,  127,   1;
  1, 36, 462, 2646, 6951, 7770, 3025, 255, 1;
  ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Noureddine Chair, Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys. 327, No. 12, 3116-3129 (2012), eq. (27).
• Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, Coefficientwise total positivity of some matrices defined by linear recurrences, arXiv:2012.03629 [math.CO], 2020.
• T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
• U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
• Eric Weisstein's World of Mathematics, Bell Polynomial
• Eric Weisstein's World of Mathematics, Empty Graph
• Eric Weisstein's World of Mathematics, Independence Polynomial
• Eric Weisstein's World of Mathematics, Sigma Polynomial
• Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

Crossrefs

See A008277 and A048993, which are the main entries for this triangle of numbers.

Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108, A213735.

Programs

Haskell

a008278 n k = a008278_tabl !! (n-1) !! (k-1)
a008278_row n = a008278_tabl !! (n-1)
a008278_tabl = iterate st2 [1] where
  st2 row = zipWith (+) ([0] ++ row') (row ++ [0])
            where row' = reverse $ zipWith (*) [1..] $ reverse row
-- Reinhard Zumkeller, Jun 22 2013

Mathematica

rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* Jean-François Alcover, Nov 17 2011, *)
Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
n = 5; Grid[Prepend[Transpose[{Range[1, n], Table[D[f[Exp[x]], {x, i}], {i, 1, n}]}], {"Order","Derivative"}], Frame -> All, Spacings -> {2, 1}] (* Leonidas Liponis, Aug 27 2024 *)

PARI

for(n=1,10,for(k=1,n,print1(stirling(n,n-k+1,2),", "))) \\ Hugo Pfoertner, Aug 30 2020

Formula

T(n, k)=0 if n < k, T(n, 0)=0, T(1, 1)=1, T(n, k) = (n-k+1)*T(n-1, k-1) + T(n-1, k) otherwise.

O.g.f. for the k-th column: 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m if k >= 2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. - Wolfdieter Lang, Oct 14 2005

E.g.f. for the k-th column (with offset n=0): E(k,x):=exp(x)*Sum_{m=0..k-1} A112493(k-1,m)*(x^(k-1+m))/(k-1+m)! if k >= 1. - Wolfdieter Lang, Oct 14 2005

a(n) = abs(A213735(n-1)). - Hugo Pfoertner, Sep 07 2020

Extensions

Name edited by Gus Wiseman, Aug 11 2020