cp's OEIS Frontend

Prime p divides a(p+1). - Alexander Adamchuk, Jul 05 2006

Also number of lists of elements from {1,..,n} with (1st element) = (smallest element), where a list means an ordered subset (cf. A000262), see also Haskell program. - Reinhard Zumkeller, Oct 26 2010

a(n+1) = p_n(-1) where p_n(x) is the unique degree-n polynomial such that p_n(k) = A133942(k) for k = 0, 1, ..., n. - Michael Somos, Apr 30 2012

a(n) = A006231(n) + n. - Geoffrey Critzer, Oct 04 2012

Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.

Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).

Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).

The reversed unsigned array is A111492.

See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=

-log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.

Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.

A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - Wolfdieter Lang, Mar 01 2014

Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - Tom Copeland, Mar 01 2014

Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - Tom Copeland, Mar 02 2014

For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - Tom Copeland, Apr 05 2014

The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - Peter Bala, Feb 13 2017

From Harry Richman, Aug 13 2024: (Start)

Euler argued this sequence converges to 0.596347... (A073003 = Gompertz's constant); see Lagarias Section 2.5.

This sequence converges in the p-adic topology, for every prime number p. (End)

We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022

The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - Gottfried Helms Apr 19 2014

From Gus Wiseman, Jan 02 2020: (Start)

Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:

{1,2,3,4} {{1},{2,3,4}} {{{1}},{{2},{3,4}}}

{{1,2},{3,4}} {{{1},{2}},{{3,4}}}

{{1,2,3},{4}} {{{1},{2,3}},{{4}}}

{{1,2,4},{3}} {{{1,2}},{{3},{4}}}

{{1,3},{2,4}} {{{1,2},{3}},{{4}}}

{{1,3,4},{2}} {{{1},{2,4}},{{3}}}

{{1,4},{2,3}} {{{1,2},{4}},{{3}}}

{{1},{2},{3,4}} {{{1}},{{3},{2,4}}}

{{1},{2,3},{4}} {{{1},{3}},{{2,4}}}

{{1,2},{3},{4}} {{{1,3}},{{2},{4}}}

{{1},{2,4},{3}} {{{1,3},{2}},{{4}}}

{{1,3},{2},{4}} {{{1},{3,4}},{{2}}}

{{1,4},{2},{3}} {{{1,3},{4}},{{2}}}

{{{1}},{{4},{2,3}}}

{{{1},{4}},{{2,3}}}

{{{1,4}},{{2},{3}}}

{{{1,4},{2}},{{3}}}

{{{1,4},{3}},{{2}}}

(End)

From Harry Richman, Mar 30 2023: (Start)

Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:

(empty) {12|3|4} {12|3|4} < {123|4}

{13|2|4} {12|3|4} < {124|3}

{14|2|3} {12|3|4} < {12|34}

{1|23|4} {13|2|4} < {123|4}

{1|24|3} {13|2|4} < {134|2}

{1|2|34} {13|2|4} < {13|24}

{123|4} {14|2|3} < {124|3}

{124|3} {14|2|3} < {134|2}

{134|2} {14|2|3} < {14|23}

{1|234} {1|23|4} < {123|4}

{12|34} {1|23|4} < {1|234}

{13|24} {1|23|4} < {14|23}

{14|23} {1|24|3} < {124|3}

{1|24|3} < {1|234}

{1|24|3} < {13|24}

{1|2|34} < {134|2}

{1|2|34} < {1|234}

{1|2|34} < {12|34}

(End)

Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - Harry Richman, Mar 30 2023

Also the exponential limit as defined in A320956 of log(x + 1).

Compare with A134685 which uses a different order with fewer entries.

For the inversion (aka reversion) of o.g.f. formal power series see A111785, and also A133437.

The sequence of row lengths of this array is p(n)=A000041(n) (number of partitions of n).

The unsigned triangle, with entries for like parts number m summed, is A134991 (2-associated Stirling numers of the second kind).

The row sums are A133942(n) = ((-1)^n) * n!, and the row sums of the unsigned array give A000311(n+1) (Schroeder's fourth problem). These sums coincide with those of the triangle A134991.

The signed a(n,k) numbers, k=1,...,p(n)=A000041(n), derive from the multinomial M_3 numbers A036040 (see also the W. Lang link there), namely, if the k-th partition of n in A-St order has exponents (enk[1],...,enk[n]) then a(n,k) = ((-1)^m)*M3(n+m, (ehatnk[1],...,ehatnk[n+m])) with m the number of parts, i.e., m:=Sum_{j=1..n} enk[j], and M3(n+m, (ehatnk[1],...,ehatnk[n+m])):=(n+m)!/(Product_{j=1..n+m} j!^ehatnk[j]*ehatnk[j]!), where the n+m exponents ehatnk are ehatnk[1]:=0, (ehatnk[2],...,ehatnk[n+1]) := (enk[1],...,enk[n]), and (ehatnk[n+1],...,ehatnk[n+m]):=(0,...,0) (i.e., m-1 zeros).

The compositional inverse of the formal power series of the e.g.f. type g(x) = Sum_{j>=1} g[j]*(x^j)/j! is f = g^[-1] with f(y) = Sum_{n>=1} f[n]*(y^n)/n!, and f[n] = fhat[n]/g[1]^(2*n-1) with fhat[1]=1 (f[1] = 1/g[1]) and f[n+1] = Sum_{k=1..p(n)} a(n,k)*g(n,k), n >= 1, where p(n) = A000041(n) (number of partitions of n), and g(n,k) is the monomial in coefficients of g(x) corresponding to the k-th partition of 2*n with n parts in A-St order. For details and a remark on the Faa di Bruno Hopf algebra see the W. Lang link.