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A047970
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Antidiagonal sums of nexus numbers (A047969).
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23
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1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486, 182905, 919146, 4866871, 27068420, 157693007, 959873708, 6091057009, 40213034874, 275699950381, 1959625294310, 14418124498211, 109655727901592, 860946822538675, 6969830450679864, 58114638923638573
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OFFSET
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0,2
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COMMENTS
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A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)
Number of ordered factorizations over the Gaussian polynomials.
Apparently, also the number of permutations in S_n avoiding {bar 3}{bar 1}542 (i.e., every occurrence of 542 is contained in an occurrence of a 31542). - Lara Pudwell, Apr 25 2008
With offset 1, apparently the number of sequences {b(m)} of length n of positive integers with b(1) = 1 and, for all m > 1, b(m) <= max{b(m-1) + 1, max{b(i) | 1 <= i <= m - 1}}. This sequence begins 1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486. The term 144 counts the length 6 sequence 1, 2, 3, 1, 1, 3, for instance. Contrast with the families of sequences discussed in Franklin T. Adams-Watters's comment in A005425. - Rick L. Shepherd, Jan 01 2015
a(n-1) for n >= 1 is the number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] with s(0)=0 and s(k) <= the number of fixed points in the prefix, see example. - Joerg Arndt, Mar 08 2015
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) = e(k). [Martinez and Savage, 2.15] - Eric M. Schmidt, Jul 17 2017
a(n) counts all positive-integer m-tuples whose maximum is n-m+2. - Mathew Englander, Feb 28 2021
a(n) counts the cyclic permutations of [n+2] that avoid the vincular pattern 12-3-4, i.e., the pattern 1234 where the 1 and 2 are required to be adjacent. - Rupert Li, Jul 27 2021
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LINKS
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FORMULA
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Formal o.g.f.: (1 - x)*( Sum_{n >= 0} x^n/(1 - (n + 2)*x) ). - Peter Bala, Jul 09 2014
O.g.f.: Sum_{n>=0} (n+1)! * x^n/(1-x)^(n+1) / Product_{k=1..n+1} (1 + k*x). - Paul D. Hanna, Jul 20 2014
O.g.f.: Sum_{n>=0} x^n / ( (1 - n*x) * (1 - (n+1)*x) ). - Paul D. Hanna, Jul 22 2014
a(n) = Sum_{m = 1..n+1} Sum_{i = 0..m-1} binomial(m,i) * (n-m+1)^i.
a(n) = 1 + Sum_{i = 0..n} i * (i+1)^(n-i). (End)
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EXAMPLE
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a(3) = 1 + 5 + 7 + 1 = 14.
G.f. A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 43*x^4 + 144*x^5 + 523*x^6 + 2048*x^7 + ...
where we have the series identity:
A(x) = (1-x)*( 1/(1-2*x) + x/(1-3*x) + x^2/(1-4*x) + x^3/(1-5*x) + x^4/(1-6*x) + x^5/(1-7*x) + x^6/(1-8*x) + ...)
is equal to
A(x) = 1/(1-x) + x/((1-x)*(1-2*x)) + x^2/((1-2*x)*(1-3*x)) + x^3/((1-3*x)*(1-4*x)) + x^4/((1-4*x)*(1-5*x)) + x^5/((1-5*x)*(1-6*x)) + x^6/((1-6*x)*(1-7*x)) + ...
and also equals
A(x) = 1/((1-x)*(1+x)) + 2!*x/((1-x)^2*(1+x)*(1+2*x)) + 3!*x^2/((1-x)^3*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
(End)
There are a(4-1)=14 length-4 RGS as in the comment (dots denote zeros):
01: [ . . . . ]
02: [ . . . 1 ]
03: [ . . 1 . ]
04: [ . . 1 1 ]
05: [ . 1 . . ]
06: [ . 1 . 1 ]
07: [ . 1 . 2 ]
08: [ . 1 1 . ]
09: [ . 1 1 1 ]
10: [ . 1 1 2 ]
11: [ . 1 2 . ]
12: [ . 1 2 1 ]
13: [ . 1 2 2 ]
14: [ . 1 2 3 ]
(End)
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MAPLE
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T := proc(n, k) option remember; local j;
if k=n then 1
elif k>n then 0
else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
fi end:
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ ((1 - x) Sum[ x^k / (1 - (k + 2) x), {k, 0, n}]), {x, 0, n}]; (* Michael Somos, Jul 09 2014 *)
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PROG
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(Sage)
T = []; n = 0
while True:
T.append(1)
yield T[0]
for k in (0..n):
T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
n += 1
{a(n)=polcoeff( sum(m=0, n, (m+1)!*x^m/(1-x)^(m+1)/prod(k=1, m+1, 1+k*x +x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
{a(n)=polcoeff( sum(m=0, n, x^m/((1-m*x)*(1-(m+1)*x +x*O(x^n)))), n)}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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Antidiagonal sums of A085388 (beginning with the second antidiagonal) and A047969.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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