The On-Line Encyclopedia of Integer Sequences (OEIS)

A084938 revision #132

A084938

Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

638

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 24, 16, 9, 4, 1, 0, 120, 64, 31, 14, 5, 1, 0, 720, 312, 126, 52, 20, 6, 1, 0, 5040, 1812, 606, 217, 80, 27, 7, 1, 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1, 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0,...] = A110654 DELTA A000007.

In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/(1-(r_2*x+s_2*x*y)/(1-(r_3*x+s_3*x*y)/(1-...(continued fraction). See also the Formula section below.

T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n >= 1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan, Jul 20 2005

T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan, Jul 20 2005

This triangle * [1,2,3,...] = A134378: (1, 2, 5, 14, 44, 158, 663, ...) = row sums of triangle A134379. - Gary W. Adamson, Oct 22 2007

Riordan array (1,x*g(x)) where g(x) is the g.f. of the factorials (n!). - Paul Barry, Sep 25 2008

Modulo 2, this sequence becomes A106344.

Eigensequence of the triangle = A165489: (1, 1, 2, 6, 23, 105, 550, 3236, ...). - Gary W. Adamson, Sep 20 2009

T(n,k) is the number of permutations of {1,2,...,n} having k cycles such that the elements of each cycle of the permutation form an interval. - Ran Pan, Nov 11 2016

The convolution triangle of the factorial numbers. - Peter Luschny, Oct 09 2022

LINKS

T. D. Noe, Rows n = 0..100 of triangle, flattened

Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), Article 09.7.6.

Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.

Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.

David Callan, A combinatorial interpretation of the eigensequence for composition, arXiv:math/0507169 [math.CO], 2005.

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

H. Fuks and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.

Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

Peter Luschny, Transformations of integer sequences.

R. J. Mathar, Properties of Deleham's Delta Transformation: OEIS A084938

FORMULA

The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:

Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0. Then P(n, k) is a homogeneous polynomial in x and y of degree n and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).

T(n, n) = 1.

T(k+1, k) = A001477(k).

T(k+2, k) = A000096(k).

T(n+1, 1) = A000142(n).

T(n+2, 2) = A003149(n).

T(n+3, 3) = A090595(n).

T(n+4, 4) = A090319(n).

T(m+n, m) = Sum_{k=0..n} A090238(n, k)*binomial(m, k).

G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.

For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.

T(n,k) = Sum_{j>=0} A075834(j)*T(n-1,k+j-1).

T(2n,n) = A287899(n). - Alois P. Heinz, Jun 02 2017

From G. C. Greubel, Nov 10 2022: (Start)

Sum_{k=0..n} T(n, k) = A051295(n).

Sum_{k=0..n} (-1)^k*T(n, k) = [n=0] - A052186(n-1)*[n>0]. (End)

EXAMPLE

From Paul Barry, Sep 25 2008: (Start)

Triangle [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,0,0,0,...] begins

0, 1;

0, 1, 1;

0, 2, 2, 1;

0, 6, 5, 3, 1;

0, 24, 16, 9, 4, 1;

0, 120, 64, 31, 14, 5, 1;

0, 720, 312, 126, 52, 20, 6, 1;

0, 5040, 1812, 606, 217, 80, 27, 7, 1;

0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1;

0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1. (End)

From Paul Barry, May 14 2009: (Start)

The production matrix is

0, 1;

0, 1, 1;

0, 1, 1, 1;

0, 2, 1, 1, 1;

0, 7, 2, 1, 1, 1;

0, 34, 7, 2, 1, 1, 1;

0, 206, 34, 7, 2, 1, 1, 1;

which is based on A075834. (End)

MAPLE

DELTA := proc(r, s, n) local T, x, y, q, P, i, j, k, t1; T := array(0..n, 0..n);

for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0, k] := 1; od: for i from 0 to n do P[i, -1] := 0; od:

for i from 1 to n do for k from 0 to n do P[i, k] := sort(expand(P[i, k-1] + q[k]*P[i-1, k+1])); od: od:

for i from 0 to n do t1 := P[i, 0]; for j from 0 to i do T[i, j] := coeff(coeff(t1, x, i-j), y, j); od: lprint( seq(T[i, j], j=0..i) ); od: end;

# To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i), i= 0..40)]; s := [seq(s4(i), i=0..40)]; DELTA(r, s, 20);

# Uses function PMatrix from A357368.

PMatrix(10, n -> factorial(n - 1)); # Peter Luschny, Oct 09 2022

MATHEMATICA

a[0, 0] = 1; a[n_, k_] := a[n, k] = Sum[j! a[n - j - 1, k - 1], {j, 0, n - 1}]; Flatten[Table[a[i, j], {i, 0, 10}, {j, 0, i}]] (* T. D. Noe, Feb 22 2012 *)

DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, _] = 1; p[_, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[Floor[Range[10]/2], Prepend[Table[0, {10}], 1], 10] (* Jean-François Alcover, Sep 12 2013, after Philippe Deléham *)

PROG

(Sage)

def delehamdelta(R, S) :

L = min(len(R), len(S)) + 1

ring = PolynomialRing(ZZ, 'x')

x = ring.gen()

A = [Rk + x*Sk for Rk, Sk in zip(R, S)]

C = [ring(0)] + [ring(1) for i in range(L)]

for k in (1..L) :

for n in range(k-1, 0, -1) :

C[n] = C[n-1] + C[n+1]*A[n-1]

yield list(C[1])