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.

User: Eric S. Egge

Eric S. Egge's wiki page.

Eric S. Egge has authored 8 sequences.

A269628 Dimension of BSym_n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 35, 57, 102, 165, 279, 444, 726, 1136, 1804, 2785, 4326, 6584, 10048, 15100, 22698, 33723, 50034, 73557, 107912, 157122, 228189, 329341, 473998, 678576, 968672, 1376402, 1950177, 2751900, 3872346, 5429166, 7591294, 10579486, 14705595, 20379419, 28172006, 38836332, 53410265, 73264431, 100271052
Offset: 0

Views

Author

Eric S. Egge, Mar 01 2016

Keywords

Comments

BSym_n is the space of homogeneous series of degree n in the variables x_1, x_{-1}, x_2, x_{-2}, ... which are invariant under the natural action of the hyperoctahedral group.
a(n) is also the number of Ferrers diagrams (in the English convention) in which some boxes contain a dot, such that the dots are left-justified in each row, and for each k, the dots in rows with k boxes form a Ferrers shape, and there are n total dots and boxes.
a(n) is also the number of partitions of n in which there are 1 + floor(k/2) different parts of "type" k for each k.

Links

Crossrefs

Cf. A275416.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add(d*ceil(
      (d+1)/2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 20 2017
  • Mathematica
    Table[SeriesCoefficient[1/Product[(1 - x^j)^Floor[(j + 2)/2], {j, 1, n}], {x, 0, n}], {n, 0, 44}] (* Michael De Vlieger, Mar 06 2016 *)
  • PARI
    N=66;  x='x+O('x^N); Vec( 1 / prod(j=1, N, (1-x^j)^floor((j+2)/2) ) ) \\ Joerg Arndt, Mar 02 2016

Formula

G.f.: 1 / (Product_{j>=1} (1-x^j)^floor((j+2)/2)).
a(n) ~ Zeta(3)^(25/72) / (A^(1/2) * Pi * 2^(3/4) * sqrt(3) * n^(61/72)) * exp(1/24 - Pi^4/(384*Zeta(3)) + Pi^2*n^(1/3) / (8*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 06 2016
a(n) = A275416(2n,n). - Alois P. Heinz, Sep 19 2017

A145846 Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 6.

Original entry on oeis.org

1, 2, 8, 47, 357, 3270, 34515, 406460, 5215829, 71677058, 1041363040, 15841778155, 250494079945, 4093630537014, 68830515423498, 1186424966652225, 20902566718237725, 375485138838707850, 6863181435514906992, 127420716337372828539, 2399321143670605041105
Offset: 0

Views

Author

Eric S. Egge, Oct 21 2008

Keywords

Programs

  • Mathematica
    Table[Sum[ Binomial[n, j]^2*((1/(n - j + 1))* Binomial[2*(n - j), n - j]/((j + 1)^2*(j + 2)))* Sum[Binomial[2*i, i]*Binomial[j + 1, i + 1]* Binomial[j + 2, i + 1], {i, 0, j}], {j, 0, n}], {n, 0, 20}]

Formula

a(n) = sum(j, 0, n, C(n,j)^2 * A000108(n-j) * A005802(j)), where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)^2*(n+3)^2*(64*n^3 + 96*n^2 - 36*n - 79)*a(n) = (2240*n^7 + 13664*n^6 + 26068*n^5 + 7303*n^4 - 27638*n^3 - 20581*n^2 + 5964*n + 5940)*a(n-1) - (n-1)^2*(16576*n^5 + 61344*n^4 + 25556*n^3 - 84501*n^2 - 46860*n - 15300)*a(n-2) + 225*(n-2)^2*(n-1)^2*(64*n^3 + 288*n^2 + 348*n + 45)*a(n-3). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 5^(2*n+13/2) / (128 * Pi^2 * n^6). - Vaclav Kotesovec, Feb 18 2015

Extensions

More terms from Vaclav Kotesovec, Feb 18 2015

A145844 Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.

Original entry on oeis.org

1, 2, 8, 46, 332, 2784, 25888, 259382, 2749244, 30449416, 349379648, 4127103776, 49954287424, 617299996928, 7765434294912, 99214734136966, 1285011754097372, 16845342401817048, 223216584359771296, 2986529546579794040, 40308007404730514096, 548337251596355725312
Offset: 0

Views

Author

Eric S. Egge, Oct 21 2008

Keywords

Examples

			a(4) = 1*1*14 + 16*1*5 + 36*2*2 + 16*5*1 + 1*14*1 = 332.

Programs

  • Mathematica
    Table[Sum[ Binomial[n, j]^2*Binomial[2*j, j]* Binomial[2*(n - j), n - j]/((n - j + 1)*(j + 1)), {j, 0, n}], {n, 0, 20}]

Formula

a(n) = sum(j=0, n, A000108(j)*A000108(n-j)*C(n, j)^2 ) where A000108(n) = Catalan(n)= (2n)!/(n!(n+1)!) and C(n, j)=n!/(k!(n-j)!).
Recurrence: (n+1)^2*(n+2)*(3*n-1)*a(n) = 2*(30*n^4 + 11*n^3 - 20*n^2 - 3*n + 6)*a(n-1) - 64*(n-1)^3*(3*n+2)*a(n-2). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 2^(4*n+3) / (Pi^(3/2) * n^(7/2)). - Vaclav Kotesovec, Feb 18 2015

Extensions

More terms from Vaclav Kotesovec, Feb 18 2015

A145845 Number of permutations of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.

Original entry on oeis.org

1, 2, 7, 34, 208, 1504, 12283, 109778, 1050820, 10614856, 111978128, 1224261856, 13792583296, 159411938560, 1883550536707, 22687603653106, 277940485660012, 3456490397570392, 43565433620294908, 555752354850506312, 7167182317486700416, 93348781597357983232, 1226830676118851157712
Offset: 0

Views

Author

Eric S. Egge, Oct 21 2008

Keywords

Programs

  • Mathematica
    Table[Sum[ Binomial[n, j]^2*(1/((j + 1)^2*(j + 2)))* Sum[Binomial[2*i, i]*Binomial[j + 1, i + 1]* Binomial[j + 2, i + 1], {i, 0, j}], {j, 0, n}], {n, 0, 20}]
  • PARI
    /* using formula given; this gives fractions! */
C=binomial;
a(n)=sum(j=0, n, C(n,j)^2 * (1/((j+1)^2*(j+2))) * sum(i=0, j, C(2*i,i)*C(j+1,i+i)*C(j+2,i+1)));
\\ Joerg Arndt, Feb 18 2015
  • PARI
    /* Using a(n) = sum(j=0, n, C(n,j)^2 * A005802(j)). */
f(n)= 2 * sum(k=0,n, binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2+2*k+1-n-2*k*n)/((k+1)^2 * (k+2) * (n-k+1)));
vector(33, N, my(n=N-1); sum(j=0,n, f(j) * C(n,j)^2 ) )
\\ Joerg Arndt, Feb 18 2015

Formula

a(n) = sum(j=0, n, C(n,j)^2 * A005802(j)).
a(n) = sum(j=0, n, C(n,j)^2 * (1/((j+1)^2 (j+2))) * sum(i=0, j, C(2*i,i) * C(j+1,i+i) * C(j+2,i+1))) where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)^3*(3*n+1)*a(n) = 2*(30*n^4 + 67*n^3 + 29*n^2 - 10*n - 8)*a(n-1) - 64*(n-1)^2*n*(3*n+4)*a(n-2). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 2^(4*n+5) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Feb 18 2015

Extensions

Added more terms, Joerg Arndt, Feb 18 2015

A145868 Number of involutions of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 7.

Original entry on oeis.org

1, 2, 6, 19, 68, 255, 1020, 4221, 18186, 80304, 364476, 1684782, 7944156, 37988379, 184406508, 905147815, 4495346570, 22527055980, 113957354940, 580759868910, 2982724210440, 15414453711930, 80177422383240, 419249099692710, 2204316120027420, 11642676960438000
Offset: 0

Views

Author

Eric S. Egge, Oct 22 2008

Keywords

Programs

  • Mathematica
    Array[Cat, 21, 0]; For[i = 0, i < 21, ++i, Cat[i] = (1/(i + 1))*Binomial[2*i, i]]; Table[Sum[ Binomial[n, j]*Binomial[n - j, Floor[(n - j)/2]]* Cat[Floor[(j + 1)/2]]*Cat[Ceiling[(j + 1)/2]], {j, 0, n}], {n, 0, 15}]

Formula

a(n) = sum(j,0,n, C(n,j)*C(n-j,floor((n-j)/2))*A000108(floor((j+1)/2)) * A000108(ceiling((j+1)/2))), where C(n,j) = n!/(j!(n-j)!) and A000108(n) = Catalan(n).
Recurrence: (n+3)*(n+4)*(n+5)*(2*n+1)*(2*n+3)*a(n) = 8*(2*n+1)*(5*n^3 + 33*n^2 + 67*n + 45)*a(n-1) + 4*(n-1)*(40*n^4 + 216*n^3 + 326*n^2 + 144*n + 45)*a(n-2) - 288*(n-2)*(n-1)*(n+1)*(2*n+5)*a(n-3) - 144*(n-3)*(n-2)*(n-1)*(2*n+3)*(2*n+5)*a(n-4). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 6^(n+7/2) / (2 * Pi^(3/2) * n^(7/2)). - Vaclav Kotesovec, Feb 18 2015

Extensions

More terms from Vaclav Kotesovec, Feb 18 2015

A145867 Number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequence of length 7.

Original entry on oeis.org

1, 2, 6, 20, 74, 292, 1214, 5252, 23468, 107672, 505048, 2413776, 11723188, 57737032, 287853518, 1450697572, 7381645844, 37884748712, 195947389208, 1020610698832, 5349968198328, 28208066576176, 149526042974008, 796520870628752, 4262367319460848
Offset: 0

Views

Author

Eric S. Egge, Oct 22 2008

Keywords

Crossrefs

Cf. A001006.

Programs

  • Mathematica
    Array[Cat, 21, 0]; For[i = 0, i < 21, ++i, Cat[i] = (1/(i + 1))*Binomial[2*i, i]]; Array[Mot, 21, 0]; For[i = 0, i < 21, ++i, Mot[i] = Sum[Binomial[i, 2*j]*Cat[j], {j, 0, Floor[i/2]}]]; Table[Sum[Binomial[n, j]*Mot[j]*Mot[n - j], {j, 0, n}], {n, 0, 15}]

Formula

a(n) = sum(j,0,n, C(n,j)*A001006(j)*A001006(n-j)), where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)*(n+4)*a(n) = 6*(n^2 + 3*n + 1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2) - 24*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 9 * 6^(n+1) / (Pi * n^3). - Vaclav Kotesovec, Feb 18 2015
E.g.f.: exp(2*x)*BesselI(1,2*x)^2/x^2. - Ilya Gutkovskiy, Sep 21 2017

Extensions

More terms from Alois P. Heinz, Feb 18 2015

A145870 Number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 8.

Original entry on oeis.org

1, 2, 6, 20, 75, 301, 1287, 5762, 26875, 129520, 642452, 3264834, 16950089, 89646090, 482012650, 2629809994, 14537429823, 81313943942, 459705628930, 2624247237560, 15113949789357, 87755911422989, 513357330465591, 3023830805847910, 17925386942479025
Offset: 0

Views

Author

Eric S. Egge, Oct 22 2008

Keywords

Comments

a(n) is also the number of involutions of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 8.

Crossrefs

Cf. A000108, A001006.

Programs

  • Mathematica
    Array[Cat, 21, 0];
For[i = 0, i < 21, ++i, Cat[i] = (1/(i + 1))*Binomial[2*i, i]];
Array[Mot, 21, 0];
For[i = 0, i < 21, ++i, Mot[i] = Sum[Binomial[i, 2*j]*Cat[j], {j, 0, Floor[i/2]}]];
Table[Sum[ Binomial[n, j]*Cat[Floor[(j + 1)/2]]*Cat[Ceiling[(j + 1)/2]]* Mot[n - j],
{j, 0, n}], {n, 0, 15}]

Formula

a(n) = Sum_{j=0..n} C(n,j) * A000108(floor((j+1)/2)) * A000108(ceiling((j+1)/2)) * A001006(n-j), where C(n,j) = n!/(j!*(n-j)!), A000108(n) = Catalan(n) and A001006(n) = Motzkin(n).
From Vaclav Kotesovec, Feb 18 2015: (Start)
Recurrence: (n+3)*(n+5)*(n+6)*(192*n^2 + 992*n + 1321)*a(n) = 4*(192*n^5 + 3392*n^4 + 21897*n^3 + 64596*n^2 + 84418*n + 35925)*a(n-1) + 2*(n-1)*(3264*n^4 + 28000*n^3 + 74185*n^2 + 47329*n - 41250)*a(n-2) - 4*(n-2)*(n-1)*(3648*n^3 + 30272*n^2 + 73819*n + 38895)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*(192*n^2 + 1376*n + 2505)*a(n-4).
a(n) ~ 7^(n+9/2) / (4 * Pi^(3/2) * n^(9/2)). (End)

Extensions

More terms from Vaclav Kotesovec, Feb 18 2015

A145847 Number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 6.

Original entry on oeis.org

1, 2, 6, 19, 67, 246, 947, 3746, 15213, 62950, 264920, 1129965, 4877215, 21262918, 93522756, 414532163, 1850047621, 8307290058, 37507875950, 170191051327, 775719275151, 3550191976022, 16309030657001, 75179696666964, 347658070586857, 1612424809368446
Offset: 0

Views

Author

Eric S. Egge, Oct 21 2008

Keywords

Links

  • Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Crossrefs

Cf. A001006, A001405.

Programs

  • Mathematica
    Table[Sum[ Binomial[n, j]*Binomial[n - j, Floor[(n - j)/2]]* Sum[Binomial[j, 2*i]*Binomial[2*i, i]/(i + 1), {i, 0, Floor[j/2]}], {j, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{j=0..n} binomial(n,j)*A001006(j)*A001405(n-j).
Recurrence: (n+2)*(n+3)*(8*n+7)*a(n) = 3*(8*n^3 + 39*n^2 + 51*n + 22)*a(n-1) + (n-1)*(104*n^2 + 155*n - 30)*a(n-2) - 15*(n-2)*(n-1)*(8*n+15)*a(n-3). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 5^(n+2) / (2*Pi*n^2). - Vaclav Kotesovec, Feb 18 2015

Extensions

More terms from Vaclav Kotesovec, Feb 18 2015

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