A143983 -id:A143983 - cp's OEIS frontend

A000994 Shifts 2 places left under binomial transform.

1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105, 1992750830574681, 13817968813639426

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Nov 02 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, a(n)/A000995(n) (e.g., 77464/63117) -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n)*(1-n*x)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Nov 02 2006
Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

A000996 Shifts 3 places left under binomial transform.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 6, 17, 44, 112, 304, 918, 3040, 10623, 38161, 140074, 528594, 2068751, 8436893, 35813251, 157448068, 713084042, 3315414747, 15805117878, 77273097114, 387692392570, 1996280632656, 10542604575130, 57034787751655, 315649657181821

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..300
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Column k=3 of A143983.

a:= proc(n) option remember; local k; if n<=2 then [1,0,0][n+1] else 1+ add(binomial(n-3,k) *a(k), k=3..n-3) fi end: seq(a(n), n=0..29); # Alois P. Heinz, Sep 05 2008

a[n_] := a[n] = If[n <= 2 , {1, 0, 0}[[n+1]], 1+Sum [Binomial[n-3, k]*a[k], {k, 3, n-3}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)

G.f. A(x) satisfies: A(x) = 1 + x^3 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

More terms from Alois P. Heinz, Sep 05 2008

A010748 Shifts 4 places right under inverse binomial transform.

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 23, 65, 165, 398, 976, 2618, 7997, 27205, 97705, 355631, 1289746, 4662069, 16971775, 63150385, 243513801, 980670052, 4121324752, 17941655332, 80143362633, 364476958473, 1680382664145, 7847729640629, 37192941056498, 179431901258459

Offset: 0

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..300
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Column k=4 of A143983 (using a different offset).

T:= proc(n,k) option remember; local j; if n T(n+4,4): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 05 2008

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

A010749 Shifts 5 places right under inverse binomial transform.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 8, 30, 94, 257, 640, 1505, 3482, 8402, 22660, 70825, 248912, 924764, 3465758, 12813670, 46470377, 165908866, 588617326, 2103688426, 7696710691, 29266242303, 116732304039, 488414436671, 2126002398180, 9511898145938, 43251315994457

Offset: 0

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..300
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Column k=5 of A143983 (using a different offset).

T:= proc(n,k) option remember; local j; if n T(n+5,5): seq(a(n), n=0..32);  # Alois P. Heinz, Sep 05 2008

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

A010750 Shifts 6 places right under inverse binomial transform.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 9, 38, 131, 387, 1025, 2512, 5834, 13152, 29805, 71858, 197325, 635938, 2311865, 8867078, 34201676, 129669923, 479650565, 1731405819, 6124746296, 21382533684, 74413732788, 261584276096, 943601869926, 3547521313455, 14045005473985

Offset: 0

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..300
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Column k=6 of A143983.

T:= proc(n, k) option remember; `if`(n T(n+6, 6): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 05 2008

T[n_, k_] := T[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*T[j, k], {j, 0, n-k}]]; a[n_] := T[n+6, 6]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *)

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A000994 Shifts 2 places left under binomial transform.

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A000996 Shifts 3 places left under binomial transform.

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A010748 Shifts 4 places right under inverse binomial transform.

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A010749 Shifts 5 places right under inverse binomial transform.

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A010750 Shifts 6 places right under inverse binomial transform.

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