OFFSET
0,6
COMMENTS
Row n contains floor(n/2) + 1 terms.
LINKS
Gheorghe Coserea, Rows n = 0..202, flattened
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.
FORMULA
From Gheorghe Coserea, Jan 29 2018: (Start)
p(n) = Sum_{k=0..floor(n/2)} T(n,k)*t^(n+1-2*k) satisfies (n+2)*p(n) = (2*n+1)*t*p(n-1) + (n-1)*(n^2-t^2)*p(n-2), n >= 2. (th. 3, (iii))
E.g.f. A(x;t) = Sum_{n>=0} p(n)*x^n/n! = ((1-x)^(-t) - (1+x)^t)/x^2. (th. 3, (i))
T(n,k) = -Stirling1(n+2, n+1-2*k)/binomial(n+2,2), where Stirling1(n,k) is defined by A048994.
n-th row polynomial R(n,x) satisfies x*R(n,x^2) = (1/2)*( P(n+1,x) - P(n+1,-x) )/ binomial(n+2,2), where P(k,x) = (1 + x)*(1 + 2*x) * ... *(1 + k*x). - Peter Bala, May 14 2023
EXAMPLE
Triangle begins:
n\k| 0 1 2 3 4 5 6
-----+---------------------------------------------------
0 | 1
1 | 1
2 | 1 1
3 | 1 5
4 | 1 15 8
5 | 1 35 84
6 | 1 70 469 180
7 | 1 126 1869 3044
8 | 1 210 5985 26060 8064
9 | 1 330 16401 152900 193248
10 | 1 495 39963 696905 2286636 604800
11 | 1 715 88803 2641925 18128396 19056960
12 | 1 1001 183183 8691683 109425316 292271616 68428800
...
Polynomials p_n(t):
p_0 = t;
p_1 = t^2;
p_2 = t^3 + t;
p_3 = t^4 + 5*t^2;
p_4 = t^5 + 15*t^3 + 8*t;
p_5 = t^6 + 35*t^4 + 84*t^2;
p_6 = t^7 + 70*t^5 + 469*t^3 + 180*t;
p_7 = t^8 + 126*t^6 + 1869*t^4 + 3044*t^2;
...
A(x;t) = t + t^2*x/1! + (t^3 + t)*x^2/2! + (t^4 + 5*t^2)*x^3/3! + ...
MATHEMATICA
T[n_, k_] := Abs[StirlingS1[n+2, n-2k+1]]/Binomial[n+2, 2];
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)
PROG
(PARI)
seq(N) = {
my(p=vector(N), t='t, v); p[1] = t^2; p[2] = t^3 + t;
for (n=3, N,
p[n] = ((2*n+1)*t*p[n-1] + (n-1)*(n^2-t^2)*p[n-2])/(n+2));
v = vector(#p, n, vector(1+n\2, k, polcoeff(p[n], n+1-2*(k-1))));
concat([[1]], v);
};
concat(seq(13))
(PARI)
N=14; x='x+O('x^(N+1));
concat(apply(p->select(a->a!=0, Vec(p)), Vec(serlaplace(((1-x)^(-t) - (1+x)^t)/x^2))))
(PARI)
T(n, k) = -stirling(n+2, n+1-2*k, 1)/binomial(n+2, 2);
concat(1, concat(vector(13, n, vector(1+n\2, k, T(n, k-1)))))
\\ Gheorghe Coserea, Jan 29 2018
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jan 21 2012
EXTENSIONS
More terms from Gheorghe Coserea, Jan 29 2018
STATUS
approved