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A084938
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Triangle of numbers T(n,k), 0<=n, 0<=k: T(n,k)= sum(j>=0) j!*T(n-j-1, k-1).
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432
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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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
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 (callan(AT)stat.wisc.edu), 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 (callan(AT)stat.wisc.edu), Jul 20 2005
This triangle * [1,2,3,...] = A134378: (1, 2, 5, 14, 44, 158, 663,...) = row sums of triangle A134379. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 22 2007
Riordan array (1,xg(x)) where g(x) is the g.f. of the factorials (n!). [From Paul Barry (pbarry(AT)wit.ie), Sep 25 2008]
Modulo 2, this sequence becomes A106344 .
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).
Eigensequence of the triangle = A165489: (1, 1, 2, 6, 23, 105, 550, 3236,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2009]
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REFERENCES
| David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
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LINKS
| David Callan, A combinatorial interpretation of the eigensequence for composition
Peter Luschny, Transformations of integer sequences.
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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(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, j>=0}A075834(j)*T(n-1,k+j-1).
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EXAMPLE
| {1}, {0, 1}, {0, 1, 1}, {0, 2, 2, 1}, {0, 6, 5, 3, 1}, {0, 24, 16, 9, 4, 1}, ...
Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 25 2008: (Start)
Triangle [0,1,1,2,2,4,4,5,5,....] DELTA [1,0,0,0,0,....] begins
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 (End)
Contribution from Paul Barry (pbarry(AT)wit.ie), 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)
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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);
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PROG
| (Sage)
def delehamdelta(R, S) :
L = min(len(R), len(S)) + 1
A = [SR(R[k] + x*S[k]) for k in range(L-1)]
C = [SR(1) for i in range(L+1)]; C[0] = SR(0)
for k in (1..L) :
for n in range(k-1, 0, -1) :
C[n] = C[n-1] + C[n+1]*A[n-1]
p = expand(C[1])
print [p.coefficient(x, n) for n in (0..k-1)]
def A084938_triangle(n) :
return delehamdelta([(i+1)//2 for i in (0..n)], [0^i for i in (0..n)])
A084938_row(10) # Peter Luschny, Jan 28 2012
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CROSSREFS
| T(k, k) = 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).
Cf. A051295 (row sums), A090238, A134378, A134379.
Diagonals : A000007, A000142, A003149, A090595, A090319 ; A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.
Cf. A165489, A165490 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2009]
Sequence in context: A110314 A152882 A130167 * A135898 A131182 A093729
Adjacent sequences: A084935 A084936 A084937 * A084939 A084940 A084941
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KEYWORD
| nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 16 2003; corrections Dec 17 2008, Dec 20 2008, Feb 05 2009
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