A134402 -id:A134402 - cp's OEIS frontend

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

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.

A134400 M * A007318, where M = triangle with (1, 1, 2, 3, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 3, 9, 9, 3, 4, 16, 24, 16, 4, 5, 25, 50, 50, 25, 5, 6, 36, 90, 120, 90, 36, 6, 7, 49, 147, 245, 245, 147, 49, 7, 8, 64, 224, 448, 560, 448, 224, 64, 8, 9, 81, 324, 756, 1134, 1134, 756, 324, 81, 9, 10, 100, 450, 1200, 2100, 2520, 2100, 1200, 450, 100, 10

Offset: 0

Gary W. Adamson, Oct 23 2007

Row sums = A134401: (1, 2, 8, 24, 64, 160, 384, ...).
Triangle T(n,k), read by rows, given by [1,1,-1,1,0,0,0,0,0,...] DELTA [1,1,-1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. A134402*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 26 2007
For n > 0, from n athletes, select a team of k players and then choose a coach who is allowed to be on the team or not. - Geoffrey Critzer, Mar 13 2010
Row sums are A036289 if first term changed to zero. Diagonal sums are A023610, starting with the 2nd diagonal. Partial sums of diagonals are A002940 if first term changed to zero. - John Molokach, Jul 06 2013
For n > 0, T(n,k) is the number of states in Sokoban puzzle with n non-obstacles cells and k boxes (see Russell and Norvig at page 157). - Stefano Spezia, Dec 03 2023

			First few rows of the triangle:
  1;
  1,  1;
  2,  4,   2;
  3,  9,   9,   3;
  4, 16,  24,  16,   4;
  5, 25,  50,  50,  25,   5;
  6, 36,  90, 120,  90,  36,  6;
  7, 49, 147, 245, 245, 147, 49, 7;
  ...

Stuart Russell and Peter Norvig, Artificial Intelligence: A Modern Approach, Fourth Edition, Hoboken: Pearson, 2021.

Wikipedia, Sokoban

Cf. A002940, A003506, A007318, A036289, A134401, A134402.

T(2n,n) give A002011(n-1) for n>=1.

with(combstruct): for n from 0 to 10 do seq(`if`(n=0, 1, n)* count( Combination(n), size=m), m=0..n) od; # Zerinvary Lajos, Apr 09 2008

Join[{1},Table[Table[n*Binomial[n, k], {k,0, n}], {n, 10}]] //Flatten (* Geoffrey Critzer, Mar 13 2010 adapted by Stefano Spezia, Dec 03 2023 *)

From Geoffrey Critzer, Mar 13 2010: (Start)
T(0,0) = 1 and T(n,k) = n * binomial(n,k) for n > 0.
E.g.f. for column k is: (x^k/k!)*exp(x)*(x+k). (End)
T(n,k) = A003506(n,k) + A003506(n,k-1). - Geoffrey Critzer, Mar 13 2010
G.f.: (1-x-x*y+x^2+x^2*y+x^2*y^2)/(1-2*x-2*x*y+x^2+2*x^2*y+x^2*y^2). - Philippe Deléham, Nov 14 2013
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,0)=T(1,1)=1, T(2,0)=T(2,2)=2, T(2,1)=4, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 14 2013
E.g.f.: 1 + exp(y*x)*exp(x)*(y*x + x). - Geoffrey Critzer, Mar 15 2015

a(55)-a(65) from Stefano Spezia, Dec 03 2023

A134403 Triangle read by rows: row n consists of (n, n, (n+1), (n+2), (n+3), ...).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 4, 5, 4, 4, 5, 6, 7, 5, 5, 6, 7, 8, 9, 6, 6, 7, 8, 9, 10, 11, 7, 7, 8, 9, 10, 11, 12, 13, 8, 8, 9, 10, 11, 12, 13, 14, 15, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 11, 12, 13, 14, 15, 16

Offset: 0

Gary W. Adamson, Oct 23 2007

Row sums = A005449: (0, 2, 7, 15, 26, 40, 57, ...).

			First few rows of the triangle:
  0;
  1, 1;
  2, 2, 3;
  3, 3, 4, 5;
  4, 4, 5, 6,  7;
  5, 5, 6, 7,  8,  9;
  6, 6, 7, 8,  9, 10, 11;
  7, 7, 8, 9, 10, 11, 12, 13;
  ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A134402, A005449.

Table[{n,n+Range[0,n-1]},{n,0,20}]//Flatten (* Harvey P. Dale, Dec 26 2020 *)

Equals (A000012 * A134402 + A134402 * A000012) - A000012.

Leading term changed from 1 to 0 by N. J. A. Sloane, Apr 06 2008

Corrected and extended by Harvey P. Dale, Dec 26 2020

cp's OEIS Frontend

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

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A134400 M * A007318, where M = triangle with (1, 1, 2, 3, ...) in the main diagonal and the rest zeros.

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A134403 Triangle read by rows: row n consists of (n, n, (n+1), (n+2), (n+3), ...).

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