A188588 -id:A188588 - cp's OEIS frontend

A188587 1-Euler triangle.

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 30, 14, 1, 1, 33, 146, 146, 33, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 151, 2241, 7687, 7687, 2241, 151, 1, 1, 310, 7780, 44194, 76870, 44194, 7780, 310, 1, 1, 629, 25820, 231236, 649514, 649514, 231236, 25820, 629, 1, 1, 1268, 83121, 1131504, 4866222, 7794168, 4866222, 1131504, 83121, 1268, 1

Offset: 0

Paul Barry, Apr 04 2011

Formed with the same recurrence as the Euler triangle A008292 (adjusted for offset), but with the middle element of row n=2 set to 1.

Row sums are A188588. Second column is A188589.

			Triangle begins
  1;
  1,   1;
  1,   1,     1;
  1,   5,     5,      1;
  1,  14,    30,     14,      1;
  1,  33,   146,    146,     33,      1;
  1,  72,   603,   1168,    603,     72,      1;
  1, 151,  2241,   7687,   7687,   2241,    151,     1;
  1, 310,  7780,  44194,  76870,  44194,   7780,   310,   1;
  1, 629, 25820, 231236, 649514, 649514, 231236, 25820, 629, 1;

A188587 := proc(n,k) if k < 0 or k > n then 0; elif k=0 or k= n or n=2 then 1; else (n-k+1)*procname(n-1,k-1)+(k+1)*procname(n-1,k) ; end if; end proc:
seq(seq(A188587(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Apr 13 2011

T(n,k) = 0 if k < 0 or k > n.
T(n,k) = 1 if k=0 or k=n or n=2.
T(n,k)= (n-k+1)*T(n-1,k-1) + (k+1)*T(n-1,k), n > 2 and 1 <= k < n.

A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 5, 3, 0, 0, 24, 28, 12, 0, 0, 334, 390, 240, 60, 0, 0, 13262, 10776, 6090, 2280, 360, 0, 0, 1106862, 615860, 255570, 92820, 23520, 2520, 0, 0, 175376048, 66625504, 19275424, 5446560, 1429680, 262080, 20160, 0, 0, 52257938968, 13210716600, 2592577512, 520122456, 112145040, 22649760, 3144960, 181440, 0, 0

Offset: 0

Andrew Howroyd, Feb 25 2024

			Triangle begins:
        1;
        1,      0;
        2,      0,      0;
        5,      3,      0,     0;
       24,     28,     12,     0,     0;
      334,    390,    240,    60,     0,    0;
    13262,  10776,   6090,  2280,   360,    0, 0;
  1106862, 615860, 255570, 92820, 23520, 2520, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Row sums are A006125.
Column k=0 is A370066.
Cf. A188588, A370064 (connected).

\\ Needs G, J defined in A370064.
T(n)={my(v=Vec( ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ), w=Vec(serlaplace(exp(sum(k=1, n, Polrev(J(v[k],k),y)*x^k, O(x*x^n)) )))); vector(#w, n, Vecrev(w[n],n))}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

Exponential transform of A370064.

T(n+2, n) = A188588(n + 1).

cp's OEIS Frontend

A188587 1-Euler triangle.

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