cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-29 of 29 results.

A128745 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the last peak equal to k (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 10, 12, 8, 21, 32, 36, 32, 16, 79, 116, 124, 112, 80, 32, 311, 448, 468, 416, 320, 192, 64, 1265, 1800, 1860, 1640, 1280, 864, 448, 128, 5275, 7440, 7640, 6720, 5280, 3712, 2240, 1024, 256, 22431, 31426, 32136, 28256, 22336, 16032, 10304
Offset: 1

Views

Author

Emeric Deutsch, Mar 31 2007

Keywords

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
Row sums yield A002212.
T(n,1) = A033321(n-1).
Sum_{k=1..n} k*T(n,k) = A128746(n).

Examples

			T(3,2)=4 because we have UDUUDD, UDUUDL, UUDUDD and UUDUDL.
Triangle starts:
   1;
   1,  2;
   2,  4,  4;
   6, 10, 12,  8;
  21, 32, 36, 32, 16;

Links

  • E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A002212, A033321, A128746.

Programs

  • Maple
    g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z/(1-2*t*z-z*g): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

Formula

G.f.: t*z/(1 - z*g - 2*t*z), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).

A187914 Generalized Riordan array based on the binomial transform of the Fine's numbers A000957.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 4, 1, 21, 36, 15, 6, 1, 79, 137, 58, 29, 7, 1, 311, 543, 232, 132, 37, 9, 1, 1265, 2219, 954, 590, 179, 57, 10, 1, 5275, 9285, 4010, 2628, 837, 315, 68, 12, 1, 22431, 39587, 17156, 11732, 3861, 1629, 396, 94, 13, 1, 96900, 171369, 74469, 52608, 17726, 8127, 2133, 612, 108, 15, 1
Offset: 0

Views

Author

Paul Barry, Mar 15 2011

Keywords

Comments

Row sums are A033321(n+1). Second column is A002212(n+1). Equal to A007318*A187913.

Examples

			Triangle begins
1,
1, 1,
2, 3, 1,
6, 10, 4, 1,
21, 36, 15, 6, 1,
79, 137, 58, 29, 7, 1,
311, 543, 232, 132, 37, 9, 1,
1265, 2219, 954, 590, 179, 57, 10, 1,
5275, 9285, 4010, 2628, 837, 315, 68, 12, 1,
22431, 39587, 17156, 11732, 3861, 1629, 396, 94, 13, 1
Production matrix is
1, 1,
1, 2, 1,
1, 2, 1, 1,
1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 1,
1, 2, 1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 2, 1, 1,
1, 2, 1, 2, 1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 2, 1, 2, 1, 1;
Hence, for instance, we have
79=1*0+1.21+1.36+1.15+1.6+1.1;
137=1.21+2.36+2.15+2.6+2.1;
58=1.36+1.15+1.6+1.1

Formula

Let g(x)=(1+x-sqrt(1-6x+5x^2))/(2x(2-x)) be the g.f. of A033321, the binomial transform of the Fine numbers.
Then the g.f. of the k-th column is x^k*g(x)^((k+2)/2)/(1-2*x*g(x))^(k/2) if k is even, and
x^k*g(x)^((k+1)/2)/(1-2*x*g(x))^((k+1)/2) if k is odd. Otherwise put, column k has g.f.
g.f. x^k*g(x)^(k+1)/(1-xg(x)-x^2g(x)^2)^floor((k+1)/2).

A214611 Number of permutations of length n sortable by a pop stack followed by a queue and then by a stack.

Original entry on oeis.org

1, 2, 6, 24, 120, 685, 4148, 25661, 159829, 997870, 6240672, 39111747, 245782289, 1549368610
Offset: 1

Views

Author

Vincent Vatter, Mar 06 2013

Keywords

Examples

			There are only a(6)=685 permutations of length 6 that can be sorted by this machine; the machine can sort all shorter permutations.

Links

Crossrefs

Cf. A033321.

Extensions

a(11)-a(14) from Bert Dobbelaere, Jun 18 2024

A262664 Expansion of (1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x).

Original entry on oeis.org

1, 1, 3, 13, 59, 271, 1257, 5881, 27715, 131395, 626033, 2995147, 14380181, 69249337, 334345091, 1617924973, 7844900339, 38105139907, 185380469961, 903147125143, 4405621159969, 21515837558557, 105188202097091, 514747668977263
Offset: 0

Views

Author

Vladimir Kruchinin, Sep 26 2015

Keywords

Links

Crossrefs

Cf. A033321.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Sum[2^i Binomial[k, n - k - i] Binomial[k + i - 1, i] (-1)^(n - k - i), {i, 0, n - k}], {k, 0, n}], {n, 0, 23}] (* Michael De Vlieger, Sep 26 2015 *)
  • Maxima
    a(n):=sum(binomial(n,k)*sum(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i),i,0,n-k),k,0,n);
  • PARI
    x='x+O('x^50); Vec((1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x)) \\ G. C. Greubel, Jun 04 2017

Formula

a(n) = Sum_{k=0..n}(binomial(n,k)*Sum_{i=0..n-k}(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))).
G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A033321.
a(n) ~ 5^(n+1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 29 2015
D-finite with recurrence: 2*n*(3*n-4)*a(n) = (39*n^2 - 70*n + 28)*a(n-1) - (48*n^2 - 103*n + 34)*a(n-2) + 5*(n-2)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Sep 29 2015

A369431 a(n) is the number of permutations of [n] which avoid the patterns 1234, 1324, 1342, and 2413.

Original entry on oeis.org

1, 1, 2, 6, 20, 66, 214, 688, 2206, 7070, 22660, 72634, 232830, 746352, 2392486, 7669286, 24584436, 78807122, 252621702, 809796400, 2595858574
Offset: 0

Views

Author

Matt Slattery-Holmes, Jan 23 2024

Keywords

Examples

			For n = 4, the valid permutations are the 20 which are not elements of the set {1234,1324,1342,2413}, hence a(4) = 20.

Links

Crossrefs

Cf. A033321 (avoiding 1234, 1324, 1342), A369626 (avoiding 1234, 1324, 2413), A053617 (avoiding 1234, 1324), A165530 (avoiding 1234 and 2413).

Extensions

a(13)-a(20) from Martin Ehrenstein, Feb 24 2024

A369626 a(n) is the number of permutations of [n] which avoid the patterns 1234, 1324, and 2413.

Original entry on oeis.org

1, 1, 2, 6, 21, 75, 265, 925, 3201, 11017, 37793, 129393, 442497, 1512225, 5165953, 17643457, 60250113, 205729921, 702452225, 2398414593, 8188884993
Offset: 0

Views

Author

Matt Slattery-Holmes, Feb 05 2024

Keywords

Examples

			All 6 of the permutations of length 3 avoid all patterns of length 4, so a(3)=6.

Crossrefs

Cf. A033321 (avoiding 1234, 1324, and 1342), A369431 (avoiding 1234, 1324, 1342, and 2413).

Extensions

a(11)-a(20) from Martin Ehrenstein, Feb 24 2024

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 6, 8, 6, 0, 1, 18, 30, 20, 10, 0, 1, 57, 108, 90, 40, 15, 0, 1, 186, 399, 378, 210, 70, 21, 0, 1, 622, 1488, 1596, 1008, 420, 112, 28, 0, 1, 2120, 5598, 6696, 4788, 2268, 756, 168, 36, 0, 1, 7338, 21200, 27990, 22320, 11970, 4536, 1260, 240, 45
Offset: 0

Views

Author

Philippe Deléham, Dec 13 2009

Keywords

Examples

			Triangle begins : 1 ; 0,1 ; 1,0,1 ; 2,3,0,1 ; 6,8,6,0,1 ; 18,30,20,10,0,1 ; ...

Crossrefs

Cf. A007318

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000957(n+1), A033321(n), A033543(n) for x = 0,1,2 respectively. Sum_{k, 0<=k<=n} T(n,k)*(-1)^(n-k)*x^k = A054341(n), A059738(n), A049027(n+1) for x = 2,3,4 respectively.

A326348 Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

Original entry on oeis.org

1, 1, 2, 6, 24, 115, 609, 3409, 19728, 116692, 701062, 4261581, 26146111, 161631115, 1005522262, 6289410686, 39525228204, 249427451071, 1579885391573, 10040587733693, 64004713573508, 409139527503760, 2622049900367018, 16843666877986873, 108438876033442579
Offset: 0

Views

Author

Robert Brignall, Sep 11 2019

Keywords

Examples

			There are a(5) = 115 permutations of length 5 which can be expressed as a juxtaposition of a separable permutation (avoiding 2413 and 3142) with an increasing permutation. These 5 cannot be expressed: 25143, 35142, 35241, 41532 and 42531.

Links

Crossrefs

Other juxtapositions of algebraic classes with monotone ones are enumerated by A033321, A165538, and A278301.

Programs

  • Mathematica
    CoefficientList[Series[(Sqrt[1 - 6*x + x^2]*(2 - 4*x + x^2)*Sqrt[1 - 8*x + 8*x^2]) / (4*(1 - x)*(-2 + 7*x - 7*x^2 + x^3)) + (-10 + 54*x - 99*x^2 + 66*x^3 - 9*x^4 + Sqrt[1 - 6*x + x^2]*(-2 + 10*x - 15*x^2 + 7*x^3) + Sqrt[1 - 8*x + 8*x^2]*(2 - 6*x + x^2 + 6*x^3 - x^4))/(4*(1 - x)^2*(-2 + 7*x - 7*x^2 + x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 07 2024 *)

Formula

G.f.: (2-4*z+z^2)*x*y/(4*(1-z)*(-2+7*z-7*z^2+z^3)) + ((-2+10*z-15*z^2+7*z^3)*x + (2-6*z+z^2+6*z^3-z^4)*y - 10+54*z-99*z^2+66*z^3-9z^4)/(4*(1-z)^2*(-2+7*z-7*z^2+z^3)) where x=sqrt(1-6*z+z^2) and y=sqrt(1-8*z+8z^2).
a(n) ~ (63 + 8*sqrt(2) + 3*sqrt(41 + 40*sqrt(2))) * 2^(3*n/2 - 1) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * (73 + 53*sqrt(2)) * n^(3/2)). - Vaclav Kotesovec, Jul 07 2024

A337522 Number of permutations of length n that are sorted to the identity by a consecutive-312-avoiding-stack followed by a classical-21-avoiding stack.

Original entry on oeis.org

1, 1, 2, 5, 15, 50, 179, 675, 2649, 10734
Offset: 0

Views

Author

Colin Defant, Aug 30 2020

Keywords

Examples

			Sending the permutation 132 through a consecutive-312-avoiding stack results in 231, and a classical 21-avoiding stack then sends 231 to 213, which is not the identity 123. Applying this procedure to any permutation of length 3 other than 132 results in 123, so a(3)=5.

Links

Crossrefs

Cf. A033321, A001006, A337495, A011782, A001405
Previous Showing 21-29 of 29 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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