A134991 -id:A134991 - cp's OEIS frontend

A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.

1, 1, 3, 1, 15, 10, 1, 105, 105, 25, 1, 945, 1260, 490, 56, 1, 10395, 17325, 9450, 1918, 119, 1, 135135, 270270, 190575, 56980, 6825, 246, 1, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1

Offset: 0

N. J. A. Sloane, Apr 05 2012

It appears that the sum of row(n) is A000311(n+1). - Michel Marcus, Feb 07 2013

Conjecture on row sums was proved in the first paragraph of the formula section of the reverse matrix A134991 in 2008 (e.g.f. evaluated at t=1). - Tom Copeland, Jan 03 2016

			Triangle begins:
      1
      1
      3     1
     15    10    1
    105   105   25    1
    945  1260  490   56   1
  10395 17325 9450 1918 119 1 ...

Charles Jordan, Calculus of Finite Differences, Chelsea 1950, p. 172, Table C_{m, i}.

L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90. See page 85.
José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See Table 8 p. 11.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. MR1696232 (2000d:11101)
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.

See A134991, which is the mirror image and is the main entry for this triangle, for further information.

Cf. A000311.

A181996 := (n,k) -> add((-1)^(n - k + m)*binomial(2*n - k, n + m)*Stirling2(n + m, m), m = 0..n-k):
seq(seq(A181996(n, k), k = 0..n-1+0^n), n=0..8); # Peter Luschny, Feb 19 2021

T(n,k) = {if (n == 0, return(1)); if (k == 0, return (prod(x=2,n, 2*x-1))); if (k == n, return (0)); return((2*n-1-k)*T(n-1,k) + (n-k)*T(n-1, k-1));} \\ Michel Marcus, Feb 07 2013

T(n, k) = Sum_{m = 0..n-k} (-1)^(n - k + m)*C(2*n - k, n + m)*Stirling2(n + m, m). - Peter Luschny, Feb 19 2021

More terms from Michel Marcus, Feb 07 2013

A176740 Inversion of e.g.f. formal power series. Partition array in Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

-1, -1, 3, -1, 10, -15, -1, 15, 10, -105, 105, -1, 21, 35, -210, -280, 1260, -945, -1, 28, 56, 35, -378, -1260, -280, 3150, 6300, -17325, 10395, -1, 36, 84, 126, -630, -2520, -1575, -2100, 6930, 34650, 15400, -51975, -138600, 270270, -135135, -1, 45, 120, 210, 126, -990, -4620, -6930, -4620, -5775

Offset: 0

Wolfdieter Lang, Jul 14 2010

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.

			  -1;
  -1,  3;
  -1, 10, -15;
  -1, 15,  10, -105,  105;
  -1, 21,  35, -210, -280, 1260, -945;
...
a(4,4): 4th partition of 4 has exponents (2,1,0,0) with m=3, and the derived exponents ehatm are (0,2,1,0,0,0,0) with one leading and 2 extra trailing zeros. (4+3)!/(2!^2*2!*3!^1*1!) = 105, hence a(4,4) = ((-1)^3)*105 = -105.
fhat[4] = -1*g[1]^2*g[4] +10*g[1]*g[2]*g[3] - 15*g[2]^3 (n=3: 3 parts partitions of 6 for the g-monomials in A-St order).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831-2.
R. Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001, p. 175, eq. (13.84).
Ch. A. Charalambides, Enumerative Combinatorics, Chapman &Hall/CRC, 2002, p. 437, eq. (11.43) with p. 428. eq. (11.29).

W. P. Johnson, Combinatorics of Higher Derivatives of Inverses, Amer. Math. Monthly 109 (3), (2002), 273-277
Wolfdieter Lang, E.g.f. Lagrange inversion partition array.

See the fhat[n] formula explained above, and the W. Lang link for more details.

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A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.

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