A025035 -id:A025035 - cp's OEIS frontend

A003190 Number of connected 2-plexes.

1, 0, 1, 3, 29, 2101, 7011181, 1788775603301, 53304526022885278403, 366299663378889804782330207902, 1171638318502622784366970315262493034215728, 3517726593606524901243694560022510194169866584119717555335

Offset: 1

N. J. A. Sloane

The Palmer reference (incorrectly) has a(7)=7011349, a(8)=1788775603133, a(9)=53304526022885278659. - Sean A. Irvine, Mar 05 2015

Also connected 3-uniform hypergraphs on n vertices. - Gus Wiseman, Feb 23 2019

			From _Gus Wiseman_, Feb 23 2019: (Start)
Non-isomorphic representatives of the a(5) = 29 2-plexes:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.

Column k=3 of A301924.

Cf. A000665 (unlabeled 3-uniform), A025035, A125791 (labeled 3-uniform), A289837, A301922, A302374 (labeled 3-uniform spanning), A302394, A306017, A319540, A320395, A322451 (unlabeled 3-uniform spanning), A323292-A323299.

Inverse Euler transform of A000665. - Sean A. Irvine, Mar 05 2015

a(7)-a(9) corrected and extended by Sean A. Irvine, Mar 05 2015

A181386 Tetrahedron of terms C(r,n,m) representing the number of ways of choosing m disjoint subsets of r members from an original set of n members.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 4, 6, 4, 1, 1, 3, 1, 1, 1, 1, 5, 10, 10, 5, 1, 1, 6, 3, 1, 1, 1, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 15, 1, 4, 1, 1, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 15, 45, 15, 1, 10, 1, 1, 1, 1, 1, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 21, 105, 105, 1, 20

Offset: 1

Frank M Jackson, Oct 16 2010

The start index for r is 1 but the start index for m and n is 0. For each value of r, the triangle T_r(n,m) has row n containing 1 + floor(n/r) terms.
From Frank M Jackson, Nov 20 2010: (Start)
C(r,mr,m) = C(r,mr-1,m-1).
C(1,m,m) = A000012, C(2,2m,m) = A001147,
C(3,3m,m), ..., C(10,10m,m) = A025035, ..., A025042.
C(2,26,10) = 150738274937250 and represents the number of possible plugboard settings for a WWII German Enigma Enciphering Machine.
C(r,2r,2) = A001700, C(r,3r,3) = A060542, C(r,4r,4) = A082368.
C(r,n,m) = C(r,mr-1,m-1)*binomial(n,rm),
  and applied recursively gives the identity
C(r,n,m) = Binomial(n,r*m) * Product_{p=1..m} Binomial(r*(m-p+1)-1,r-1).
(End)
C(2,26,10) = A266365(10), where 26 is the size of the alphabet. - Jonathan Sondow, Dec 29 2015

			r=1, C(1,n,m) is
  1
  1, 1
  1, 2,  1
  1, 3,  3,  1
  1, 4,  6,  4, 1
  1, 5, 10, 10, 5, 1
r=2, C(2,n,m) is
  1
  1
  1,  1
  1,  3
  1,  6,  3
  1, 10, 15
r=3, C(3,n,m) is
  1
  1
  1
  1,  1
  1,  4
  1, 10

C(1,n,m) = T_1(n,m) = A007318, C(2,n,m) = T_2(n,m) = A100861, and C(2,26,m) = A266365.

Flatten[Table[{n!/((n-r*m)!*m!*r!^m)}, {r, 1, 50}, {n, 0, 50}, {m, 0, Floor[n/r]}]]

C(r,n,m) = n!/((n-r*m)!*m!*(r!)^m).

A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.

Original entry on oeis.org

1, 0, 1, 7, 3, 0, 219, 56, 5, 0, 12861, 2352, 183, 4, 0, 1215794, 174137, 11145, 323, 1, 0, 169509845, 19970411, 1078977, 30833, 334, 0, 0, 32774737463, 3280250014, 153076174, 4056764, 55379, 206, 0, 0, 8400108766161, 730845033406, 29989041076, 727278456, 10341101, 67730, 70, 0, 0

Offset: 0

Donovan Young, May 26 2020

Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 3n, see [Young].

			Triangle begins:
      1;
      0,    1;
      7,    3,   0;
    219,   56,   5, 0;
  12861, 2352, 183, 4, 0;
  ...
For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A025035.
Column k=0 is column 0 of A334056.
Cf. A079267, A334056, A334057, A334058, A334059, A325753.

CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]

T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }

G.f.: Sum_{j>=0} (3*j)! * y^j * (1-(1-z)*y)^(3*j+1) / (j! * 6^j * (1-(1-z)*y^2)^(3*j+1)).

A211310 a(n) = number |fdw(P,(n))| of entangled P-words with s=3.

Original entry on oeis.org

1, 18, 1566, 354456, 163932120, 134973740880, 180430456454640, 366311352681348480

Offset: 1

N. J. A. Sloane, Apr 08 2012

nonn

See Jenca and Sarkoci for the precise definition.

Gejza Jenca and Peter Sarkoci, Linear extensions and order-preserving poset partitions, arXiv preprint arXiv:1112.5782, 2011

Cf. A014606, A025035.

From Peter Bala, Sep 05 2012: (Start)
Conjectural e.g.f.: 2 - 1/A(x), where A(x) = sum {n = 0..inf} (3*n)!/6^n*x^n/n! is the e.g.f. for A014606 (also the o.g.f. for A025035).
If true, this leads to the recurrence equation: a(n) = (3*n)!/6^n - sum {k = 1..n-1} (3*k)!/6^k*binomial(n,k)*a(n-k) with a(1) = 1.
(End)

A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

Original entry on oeis.org

1, 7, 3, 127, 123, 30, 4369, 6822, 3579, 630, 243649, 532542, 439899, 162630, 22680, 20036983, 56717781, 64697499, 37155267, 10735470, 1247400, 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200, 343141433761, 1427877062076, 2563294235106, 2572662311496, 1558544277681, 569674791180, 116270210700, 10216206000

Offset: 1

Matthew Miller, Dec 11 2018

It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.
It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.
It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.
It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.
This triangular array arises as the coefficients from terms in the inverse expansion of the function f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t evaluated at t = 0 for even values of the operation, using a method of Dominici's (nested derivatives, referenced below).
Without proof, appears to be related to computing the 'critical t-value' of Student's t-distribution. (conj.) Critical t-value t_(v, beta) is equal to: sqrt((v/(1-S^2)) - v) where S = (1/2)*Sum_{k>=1} (D^(2*k-2)[f](0)*(1/(2*k-1)!)*(B(1/2, v/2)*(1-2*beta))^(2*k-1)); where (1 - beta) is the confidence interval 'atta' (for a one-tailed distribution such that 'cumulative probability' = t_atta, where beta = 1-atta), x = 1 - (v/2), v: degrees of freedom, B(1/2, v/2) = gamma(1/2)*gamma(v/2)/gamma(1/2 + v/2), D^(2*k - 2)[f](0) is a polynomial function of 'x' whose coefficients are the terms of this sequence as computed using a method of Dominici's on f(t,x) with respect to t (referenced below).

			Given D^k[f]_(b) = (d/dt [f(t)*D^(k-1)[f](t)])_t = b where D^0[f](b) = 1, then for f(t,x) = (1/2)*(1 - t^2)^(-x) where f(0) = 1/2 one obtains: D^2[f]_(0) = -x/2, D^4[f]_(0) = (x/4)*(7*x - 3), D^6[f]_(0) = -(x/8)*(127*x^2 - 123*x + 30), etc., where b is an arbitrary constant.
Triangle begins:
           1;
           7,          3;
         127,        123,          30;
        4369,       6822,        3579,        630;
      243649,     532542,      439899,     162630,      22680;
    20036983,   56717781,    64697499,   37155267,   10735470,   1247400;
  2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
         ...

Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
Wikipedia, Student's t-distribution

Cf. A002067, A007019, A025035, A002105.

cp's OEIS Frontend

A003190 Number of connected 2-plexes.

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A181386 Tetrahedron of terms C(r,n,m) representing the number of ways of choosing m disjoint subsets of r members from an original set of n members.

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A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.

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A211310 a(n) = number |fdw(P,(n))| of entangled P-words with s=3.

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A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

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Author

Keywords

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Examples

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Crossrefs