A130460 -id:A130460 - cp's OEIS frontend

A130461 Triangle, antidiagonals of an array generated from A130460.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 4, 1, 1, 1, 2, 6, 12, 5, 1, 1, 1, 2, 6, 24, 20, 6, 1, 1, 1, 2, 6, 24, 60, 30, 7, 1, 1, 1, 2, 6, 24, 120, 120, 42, 8, 1, 1, 1, 2, 6, 24, 120, 360, 210, 56, 9, 1, 1, 1, 2, 6, 24, 120, 720, 840, 336, 72, 10, 1, 1, 1, 2, 6, 24, 120, 720, 2520

Offset: 0

Gary W. Adamson, May 28 2007

Rows tend to the factorials: (1, 1, 2, 6, 24, ...). Row sums = A130476: (1, 2, 3, 5, 8, 15, 28, 61, 132, ...).

			The array =
  1, 1, 1, 1,  1,   1, ...
  1, 1, 2, 3,  4,   5, ...
  1, 1, 2, 6, 12,  20, ...
  1, 1, 2, 6, 24,  60, ...
  1, 1, 2, 6, 24, 120, ...
  1, 1, 2, 6, 24, 120, ...
  ...
First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2, 1;
  1, 1, 2, 3,  1;
  1, 1, 2, 6,  4,  1;
  1, 1, 2, 6, 12,  5,  1;
  1, 1, 2, 6, 24, 20,  6, 1;
  1, 1, 2, 6, 24, 60, 30, 7, 1;
  ...

Cf. A130460, A130476, A130477, A130478.

Let A130460 = M, an infinite lower triangular matrix and V = [1, 1, 1, ...], the first row of an array. Perform M * V = second row, ...; (n+1)-th row = M * n-th row. The triangle = antidiagonals of the array.

a(23) and a(38) corrected by Gionata Neri, Jun 22 2016

A218272 Infinitesimal generator for transpose of the Pascal matrix A007318 (as upper triangular matrices).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0

Offset: 0

Tom Copeland, Oct 24 2012

T is the transpose of A132440.
Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Then M(1) = the transpose of the lower triangular Pascal matrix A007318, with inverse M(-1).
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x) = n * P_(n-1)(x) and
  R P_n(x) = P_(n+1)(x), the matrix T represents the action of L in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1).
See A132440 as an analog and more general discussion.
Sum_{n>=0} c_n T^n / n! = e^(c.T) gives the Maurer-Cartan form matrix for the one-dimensional Leibniz group defined by multiplication of a Taylor series by the formal Taylor series e^(c.x) (cf. Olver). - Tom Copeland, Nov 05 2015
From Tom Copeland, Jul 02 2018: (Start)
The transpose Psc^Trn of the lower triangular Pascal matrix Psc = A007318 gives the numerical coefficients of the Maurer-Cartan form matrix M of the Leibniz group Leibniz(n)(1,1) presented on p. 9 of the Olver paper. M = exp[c. * T] with (c.)^n = c_n and T the Lie infinitesimal generator of this entry. The columns e^T are the rows of the Pascal matrix A007318.
M can be obtained by multiplying each n-th column vector of Psc by c_n and then transposing the result; i.e., with the diagonal matrix H = Diag(c_0, c_1, c_2, ...), M = (Psc * H)^Trn = H * Psc^Trn.
M is a matrix representation of the differential operator S = e^{c.*D} with D = d/dx, which acting on x^m gives the Appell polynomial p_m(x) = (c. + x)^m, with (c.)^k = c_k an arbitrary indeterminate except for c_0 = 1. For example, S x^2 = (c. + x)^2 = c_0*x^2 + 2*c_1*x + c_2, and M * (0,0,1,0,0,...)^Trn = (c_2,2*c_1,c_0,0,0,...)^Trn = V, so V^Trn = (0,0,1,0,...) * M^Trn = (0,0,1,0,...) * Psc * H = (c_2,2*c_1,c_0,0,...).
The differential lowering and raising operators for the Appell sequence are given by L = D and R = x + dlog(S)/dD, with L p_n(x = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x).
(End)

			Matrix T begins
  0,1;
  0,0,2;
  0,0,0,3;
  0,0,0,0,4;
  0,0,0,0,0,5;
  0,0,0,0,0,0,6;
  ...

Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
P. Olver, The canonical contact form.

Essentially the same as A134402, A132440 and A130460.

Table[PadLeft[{n+1}, n+2], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):
  1) b(n) = (n+1) * a(n+1),
  2) B(x) = D A(x), or
  3) EB(x) = DxD  EA(x),
  where D is the derivative w.r.t. x.
So the exponentiated operator can be characterized as
  4) exp(t*T) A(x) = exp(t*D) A(x) = A(x+t),
  5) exp(t*T) EA(x) = exp(t*DxD) EA(x) = exp[x*a/(1+t*a)]/(1+t*a),
     = Sum_{n>=0} (1+t*a)^(-n-1) (x*a)^n/n!, where umbrally
     a^n *(1+t*a)^(-n-1) = Sum_{j>0} binomial(n+j,j)a(n+j)t^j,
  6) exp(t*T) EA(x) = Sum_{n>=0} a(n) t^n Lag(n,-x/t),
     where Lag(n,x) are the Laguerre polynomials (A021009), or
  7) [exp(t*T) * a]_n = [M(t) * a]_n
     = Sum_{j>=0} binomial(n+j,j)a(n+j)t^j.
For more on the operator DxD, see A021009 and references in A132440.

A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 12, 1, 4, 15, 40, 60, 1, 5, 24, 90, 240, 360, 1, 6, 35, 168, 630, 1680, 2520, 1, 7, 48, 280, 1344, 5040, 13440, 20160, 1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440, 1, 9, 80, 630, 4320, 25200, 120960, 453600, 1209600, 1814400, 1, 10, 99, 880, 6930, 47520, 277200, 1330560, 4989600, 13305600, 19958400

Offset: 0

Gary W. Adamson, May 28 2007

Previous name was: Triangle generated from finite differences of A130461.
Decomposition of the permutations.
From Liam Solus, Aug 23 2018: (Start)
For k>0, T(n,k) equals the number of permutations p of [n] such that the largest index i for which p(i)>p(i+1) is k; i.e., T(n,k) is the number of permutations of [n] with maximum descent being k. See Lemma 3.4 of the paper by L. Solus below.
When T(n,k) is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h^*-polynomial is the n-th Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.
(End)

			First few rows of the triangle A130461 = (1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 2, 3, 1; 1, 1, 2, 6, 4, 1;...). Deleting the left border and taking finite differences at the top of each remaining column, we get the first few rows of this triangle:
1;
1, 1;
1, 2,  3;
1, 3,  8,  12;
1, 4, 15,  40,  60;
1, 5, 24,  90, 240,  360;
1, 6, 35, 168, 630, 1680, 2520;
...

L. Solus, Simplices for numeral systems, Transactions of the American Mathematical Society. DOI: https://doi.org/10.1090/tran/7424 (2017).

The triangle with each row reversed is A092582.
Cf. A000142 (row sums), A001710 (main diagonal), A008292.
Cf. A130460, A130461, A130476, A130478.

T := (n,k) -> (n-k+1+0^k)*((n+1)!/(n-k+2)!):
seq(seq(T(n,k),k=0..n),n=0..10); # Peter Luschny, Sep 17 2018

Flatten[Table[Table[(n - k + 1 + 0^k)*(n + 1)!/(n - k + 2)!, {k,0,n}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)

{T(n, k) = if( n<1 || k>n, 0, k==1, 1, n! * (n+1-k) / (n+2-k)!)}; /* Michael Somos, Jun 25 2017 */

Each term in n-th row divides n!.
Given triangle A130461 and deleting the left border (1,1,1,...) take finite differences by columns and reorient into rows.
T(n,k) = (n-k+1+0^k)*((n+1)!/(n-k+2)!) - Olivier Gérard, Aug 04 2012

New name using a comment by Liam Solus, Peter Luschny, Sep 17 2018

A130476 Row sums of triangle A130461.

Original entry on oeis.org

1, 2, 3, 5, 8, 15, 28, 61, 132, 325, 790, 2133, 5680, 16501

Offset: 1

Gary W. Adamson, May 28 2007

nonn

			a(6) = 15 = (1 + 1 + 2 + 6 + 4 + 1), sum of row 6 terms in triangle A130461.

Cf. A130460, A130461, A130477, A130478.

A134402 Triangle read by rows, for n > 0, n zeros followed by n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 0

Gary W. Adamson, Oct 23 2007

Multiplied by the vector [1, 2, 3, ...] from the right gives (1, 2, 6, 12, 20, 30, 42, ...), A002378.

Triangle T(n,k), read by rows, given by [0,0,0,0,0,0,0,...] DELTA [1,1,-1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 26 2007

			First few rows of the triangle:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 3;
  0, 0, 0, 0, 4;
  0, 0, 0, 0, 0, 5;
  ...

Cf. A134403, A005449.

Essentially the same as A130460.

Join[{1},Flatten[Table[PadLeft[{n},n+1,0],{n,15}]]] (* Harvey P. Dale, May 08 2012 *)

Triangle read by rows, a(0) = 1, then for n > 0, n zeros followed by n. Infinite lower triangular matrix with (1, 1, 2, 3, 4, ...) in the main diagonal and the rest zeros.

G.f.: (1-x*y+x^2*y^2)/(-1+x*y)^2. - R. J. Mathar, Aug 11 2015

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A130461 Triangle, antidiagonals of an array generated from A130460.

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A218272 Infinitesimal generator for transpose of the Pascal matrix A007318 (as upper triangular matrices).

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A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

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A130476 Row sums of triangle A130461.

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A134402 Triangle read by rows, for n > 0, n zeros followed by n.

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