A115179 -id:A115179 - cp's OEIS frontend

A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

1, 0, 1, 0, 1, 2, 0, 0, 4, 5, 0, 0, 2, 15, 14, 0, 0, 0, 15, 56, 42, 0, 0, 0, 5, 84, 210, 132, 0, 0, 0, 0, 56, 420, 792, 429, 0, 0, 0, 0, 14, 420, 1980, 3003, 1430, 0, 0, 0, 0, 0, 210, 2640, 9009, 11440, 4862, 0, 0, 0, 0, 0, 42, 1980, 15015, 40040, 43758, 16796, 0, 0, 0, 0, 0, 0, 792, 15015, 80080, 175032, 167960, 58786

Offset: 0

Paul Barry, Mar 14 2006

			Triangle begins as:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 4,  5;
  0, 0, 2, 15, 14;
  0, 0, 0, 15, 56,  42;
  0, 0, 0,  5, 84, 210, 132;
  0, 0, 0,  0, 56, 420, 792, 429;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A052709, A115178, A115178.

[Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021

Table[CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 31 2021 *)

flatten([[binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021

T(n, k) = binomial(k, n-k)*Catalan(k).
Sum_{k=0..n} T(n, k) = A052709(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115178(n) (upward diagonal sums).
T(n, k) = (-1)^(n+k)*A115179(n, k).

A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

Original entry on oeis.org

1, 4, 2, 17, 9, 3, 72, 38, 5, 12, 305, 161, 8, 51, 6, 1292, 682, 13, 216, 10, 7, 5473, 2889, 21, 915, 16, 30, 14, 23184, 12238, 34, 3876, 26, 127, 59, 25, 98209, 51841, 55, 16419, 42, 538, 250, 106, 11, 416020, 219602, 89, 69552, 68, 2279, 1059, 449, 18, 33

Offset: 1

Clark Kimberling, Oct 26 2019

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.  The appearance of a sequence s(r) below means that corresponding row of R is the same as s(r) except possibly for one or more initial terms of s(r).
Row 1 of R: A001076
Row 2 of R: A001077
Row 3 of R: A000045
Row 4 of R: A115179
Row 5 of R: A006355
Row 6 of R: A097924
Row 8 of R: A048875
Row 9 of R: A000032

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,4,17,72,...).
_______________
Northwest corner of R:
   1   4   17   72  305  1292   5473
   2   9   38  161  682  2889  12238
   3   5    8   13   21    34     55
  12  51  216  915 3876 16419  69552
   6  10   16   26   42    68    110
   7  30  127  538 2279  9654  40895

Cf. A035513, A001076, A001077, A000045, A115179, A006355, A097924, A048875, A000032, A328696, A328697.

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 0, w[n, k]/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
Table[u[n], {n, 1, 10}] (* A328695 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328695 sequence *)

A115399 Expansion of c(x^2-x^3), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, -1, 1, 2, -4, -3, 15, -1, -51, 42, 154, -274, -372, 1341, 405, -5599, 2783, 20295, -27313, -61139, 160797, 124462, -767702, 94964, 3145986, -2930086, -10956466, 21290868, 29513156, -114974111, -33473459, 522016279, -279684579, -2031228003, 2867285041, 6493289367, -17791297065

Offset: 0

Paul Barry, Mar 14 2006

Diagonal sums of A115179.

Cf. A115178.

a(n)=sum{k=0..floor(n/2), (-1)^(n-k)*C(k)C(k,n-2k)}.

Conjecture: (n+2)*a(n) +(-n-2)*a(n-1) +4*(n-1)*a(n-2) +2*(-4*n+7)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Feb 20 2015

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A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

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A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

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A115399 Expansion of c(x^2-x^3), c(x) the g.f. of A000108.

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