A000366 -id:A000366 - cp's OEIS frontend

A239895 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k components.

1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 129, 110, 45, 10, 1, 1438, 1104, 435, 105, 15, 1, 20955, 14455, 5334, 1295, 210, 21, 1, 384226, 238536, 81256, 19089, 3220, 378, 28, 1, 8623101, 4834854, 1509246, 335496, 56259, 7056, 630, 36, 1

Offset: 1

N. J. A. Sloane, Apr 04 2014

The Bell transform of A218827(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 17 2016

			Triangle begins:
       1;
       1,      1;
       3,      3,     1;
      16,     15,     6,     1;
     129,    110,    45,    10,    1;
    1438,   1104,   435,   105,   15,   1;
   20955,  14455,  5334,  1295,  210,  21,  1;
  384226, 238536, 81256, 19089, 3220, 378, 28, 1;

Kreweras, G. and Dumont, D., Sur les anagrammes alternés. (French) [On alternating anagrams] Discrete Math. 211 (2000), no. 1-3, 103--110. MR1735352 (2000h:05013).

Row sums are A000366. First column is A218827.

m = 10(*terms of A218827 for m-1 rows*); matc = Array[0&, {m, m}];
(* The function BellMatrix is defined in A264428.*)
a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
ci[n_, k_] := ci[n, k] = Module[{v}, If[matc[[n, k]] == 0, If[n == k, v = 1, If[k == 1, v = c[n], v = Sum[Binomial[n - 1, i - 1]*c[i]*ci[n - i, k - 1], {i, 1, n - k + 1}]]]; matc[[n, k]] = v]; Return[matc[[n, k]] ]];
c[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]]
T = Rest /@ BellMatrix[c[# + 1]&, m] // Rest;
Table[T[[n, k]], {n, 1, m - 1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 03 2019 *)

# uses[bell_matrix from A264428, A218827]
# Adds a column 1,0,0,0,... at the left side of the triangle.
A239895_generator = lambda n: A218827(n+1)
bell_matrix(A239895_generator, 9) # Peter Luschny, Jan 17 2016

T(n,k) = C(n-1,0)*c(1)*T(n-1,k-1) + C(n-1,1)*c(2)*T(n-2,k-1) + ... + C(n-1,n-1)*c(n-k+1)*T(k-1,k-1), where c(i) = A218827(i).

More terms from Peter Luschny, Jan 17 2016

A108959 Triangle arising in connection with deformations of type D Kleinian singularities.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 14, 7, 5, 20, 54, 76, 38, 6, 35, 154, 419, 590, 295, 7, 56, 364, 1616, 4400, 6196, 3098, 8, 84, 756, 4962, 22048, 60036, 84542, 42271, 9, 120, 1428, 12984, 85300, 379052, 1032154, 1453468, 726734, 10, 165, 2508, 30162, 274516, 1803638, 8014990, 21824737, 30733358, 15366679

Offset: 0

Paul Boddington, Jul 22 2005

			Triangle begins:
  1;
  2,  1;
  3,  4,  2;
  4, 10, 14,  7;
  5, 20, 54, 76, 38;
  ...

Paul Boddington, No-cycle algebras and representation theory, Ph.D. thesis, University of Warwick, 2004.

This sequence is an improved version of A097418. Coefficients of 1 give A000366.

Cf. A128813 (the p_k polynomials).

tabl(nn) = my(v = vector(nn)); for (n=1, nn, my(p=prod(i=1, n, x+i*(i-1)/2), q=n*p/x); v[n] = q - sum(i=1, n-1, polcoeff(p, i)*v[i])); vector(nn, k, Vec(v[k])); \\ Michel Marcus, Mar 18 2023

For k>=0 define p_k(x) = x(x+1)(x+3)...(x+k(k-1)/2) and consider the linear map taking each p_k(x) to k*p_k(x)/x. Then the images of x, x^2, x^3, ... are given by the rows. E.g., x^3 goes to 3x^2 + 4x + 2.

More terms from Michel Marcus, Mar 18 2023

A343198 Regular triangle T(n,k) of Dellac configurations with boundaries, n>=1 and k>=0.

Original entry on oeis.org

1, 2, 3, 7, 9, 15, 38, 45, 63, 111, 295, 333, 423, 621, 1131, 3098, 3393, 4059, 5373, 8127, 15123, 42271, 45369, 52155, 64665, 87939, 135729, 256335, 726734, 769005, 859743, 1019601, 1295163, 1794825, 2810403, 5364471, 15366679, 16093413, 17631423, 20256021, 24549831, 31731453, 44583183, 70558101, 135751731

Offset: 1

Michel Marcus, Apr 07 2021

			Triangle begins:
    1
    2   3
    7   9  15
   38  45  63 111
  295 333 423 621 1131
  ...

Keiichi Shigechi, Generalized Dellac configurations, arXiv:2104.02455 [math.CO], 2021. See p. 24.

Cf. A000366 (1st column), A130168 (right diagonal).

T(n, 0) = 2*T(n-1, 0) + Sum_{k=1..n-2} T(n-1, k).

cp's OEIS Frontend

A239895 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k components.

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A108959 Triangle arising in connection with deformations of type D Kleinian singularities.

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A343198 Regular triangle T(n,k) of Dellac configurations with boundaries, n>=1 and k>=0.

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