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A093375 Array T read by antidiagonals: T(k,n) = k*binomial(n+k-2, n-1). 7
1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 18, 8, 1, 6, 25, 40, 30, 10, 1, 7, 36, 75, 80, 45, 12, 1, 8, 49, 126, 175, 140, 63, 14, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 11, 100, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of n-long k-ary words avoiding the pattern 1-1'2'.

T(n,n+1) = Sum_{i=1..n} T(n,i).

Exponential Riordan array [(1+x)e^x, x] as a number triangle. - Paul Barry, Feb 17 2009

From Peter Bala, Jul 22 2014: (Start)

Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/

having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A059298. (End)

LINKS

Muniru A Asiru, Antidiagonals, n=1..100 flattened

S. Kitaev and T. Mansour, Partially ordered generalized patterns and k-ary words, arXiv:math/0210023 [math.CO], 2002.

Wikipedia, Sheffer sequence

FORMULA

Triangle = P*M, the binomial transform of the infinite bidiagonal matrix M with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal, and zeros elsewhere. P = Pascal's triangle as an infinite lower triangular matrix. - Gary W. Adamson, Nov 05 2006

From Peter Bala, Sep 20 2012: (Start)

E.g.f. for triangle: (1+z)*exp((1+x)*z) = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2/2! + ....

O.g.f. for triangle: (1-x*z)/(1-z-x*z)^2 = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2 + ....

The n-th row polynomial R(n,x) of the triangle equals (1+x)^n + n*(1+x)^(n-1) for n >= 0 and satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x+y) = sum {k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.

Matrix inverse of the triangle is a signed version of A073107. (End)

From Tom Copeland, Oct 20 2015: (Start)

With offset 0 and D = d/dx, the raising operator for the signed row polynomials P(n,x) is RP = x - d{log[e^D/(1-D)]}/dD = x - 1 - 1/(1-D) =  x - 2 - D - D^2 + ..., i.e., RP P(n,x) = P(n+1,x).

The e.g.f. for the signed array is (1-t) e^(-t) e^(x*t).

From the Appell formalism, the row polynomials PI(n,x) of A073107 are the umbral inverse of this entry's row polynomials; that is, P(n,PI(.,x)) = x^n = PI(n,P(.,x)) under umbral composition. (End)

EXAMPLE

   1   1   1   1   1   1

   2   4   6   8  10  12

   3   9  18  30  45  63

   4  16  40  80 140 224

   5  25  75 175 350 630

Triangle begins

.n\k.|....0....1....2....3....4....5....6

= = = = = = = = = = = = = = = = = = = = =

..0..|....1

..1..|....2....1

..2..|....3....4....1

..3..|....4....9....6....1

..4..|....5...16...18....8....1

..5..|....6...25...40...30...10....1

..6..|....7...36...75...80...45...12....1

...

MATHEMATICA

nmax = 10;

T = Transpose[CoefficientList[# + O[z]^(nmax+1), z]& /@ CoefficientList[(1 - x z)/(1 - z - x z)^2 + O[x]^(nmax+1), x]];

row[n_] := T[[n+1, 1 ;; n+1]];

Table[row[n], {n, 0, nmax}] // Flatten (* Jean-Fran├žois Alcover, Aug 07 2018 *)

PROG

(GAP) nmax:=14;; T:=List([1..nmax], n->List([1..nmax], k->k*Binomial(n+k-2, n-1)));;

b:=List([2..nmax], n->OrderedPartitions(n, 2));;

a:=Flat(List([1..Length(b)], i->List([1..Length(b[i])], j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Aug 07 2018

CROSSREFS

Rows include A045943. Columns include A002411, A027810.

Main diagonal is A037965. Subdiagonals include A002457.

Antidiagonal sums are A001792.

See A103283 for a signed version.

Cf. A103406, A059298, A073107 (unsigned inverse).

Sequence in context: A297224 A180383 A133807 * A103283 A104698 A067066

Adjacent sequences:  A093372 A093373 A093374 * A093376 A093377 A093378

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Apr 28 2004

STATUS

approved

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Last modified March 18 22:10 EDT 2019. Contains 321305 sequences. (Running on oeis4.)