A215002 -id:A215002 - cp's OEIS frontend

A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 6, 2, 5, 25, 25, 5, 14, 98, 174, 98, 14, 42, 378, 962, 962, 378, 42, 132, 1452, 4804, 7020, 4804, 1452, 132, 429, 5577, 22689, 43573, 43573, 22689, 5577, 429, 1430, 21450, 103510, 245962, 325590, 245962, 103510, 21450, 1430

Offset: 0

Alois P. Heinz, Jul 28 2012

T(n,k) is odd if and only if n = 2^i-1 for i in {0, 1, 2, ... } = A001477.

			Triangle T(n,k) begins:
    1;
    1,    1;
    2,    6,    2;
    5,   25,   25,    5;
   14,   98,  174,   98,   14;
   42,  378,  962,  962,  378,   42;
  132, 1452, 4804, 7020, 4804, 1452, 132;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns 0-5 give: A000108, A214955, A215298, A215299, A215300, A215301.
Row sums give: A215002.
Central row elements give: A214801.

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
      `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
    end:
T:= (n, k)-> b(n, k, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := b[n, k, n-k]; Table[T[n, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

@CachedFunction
def B(x, y, z) :
    if z > y : return B(x, z, y)
    if x==y and y==z and z==0 : return 1
    a = B(x-1, y, z) if x>y and x>z else 0
    b = B(x, y-1, z) if y>0 else 0
    c = B(x, y, z-1) if z>0 else 0
    return a + b + c
T = lambda n, k: B(n, k, n-k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# After Maple code of Alois P. Heinz. Peter Luschny, Jul 30 2012

A214955 Number of solid standard Young tableaux of shape [[n,n-1],[1]].

Original entry on oeis.org

1, 6, 25, 98, 378, 1452, 5577, 21450, 82654, 319124, 1234506, 4784276, 18572500, 72209880, 281150505, 1096087770, 4278278070, 16717354500, 65388738030, 256000696380, 1003116947820, 3933750236520, 15437682614250, 60625494924228, 238235373671148, 936735006679752

Offset: 1

Alois P. Heinz, Jul 30 2012

nonn

a(n) is odd if and only if n = 2^i-1 for i in {1, 2, 3, ...} = A000027.

Form an array with m(1,n) = n*(n+1)/2, m(n,1) = n*(n-1)+1, and m(i,j) = m(i,j-1) + m(i-1,j); A000217 in the top row, A002061 in the first column, A086514 in the second column. Then on the diagonal m(n,n) = a(n). - J. M. Bergot, May 02 2013

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012.
Wikipedia, Young tableau.

Column k=1 of A214775.

Cf. A000027, A000108, A006752, A060747, A215002.

a:= proc(n) option remember;
      `if`(n<2, n, 2*(2*n-1)^2*a(n-1)/((n+1)*(2*n-3)))
    end:
seq(a(n), n=1..30);

a[n_]:= a[n] = If[n<2, n, 2*(2*n-1)^2*a[n-1]/((n+1)*(2*n-3))]; Array[a, 30] (* Jean-François Alcover, Aug 14 2017, translated from Maple *)

a(n) = (2*n-1) * binomial(2*n,n)/(n+1); \\ Michel Marcus, Mar 06 2022

a(n) = 2*(2*n-1)^2/((n+1)*(2*n-3)) * a(n-1) for n>1; a(1) = 1.
a(n) = (2*n-1) * C(2*n,n)/(n+1) = A060747(n) * A000108(n).
a(n) = [x^n] x*(1 + 2*x)/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 12 2017
Sum_{n>=1} 1/a(n) = 1/6 + G + 13*Pi/(36*sqrt(3)) - Pi*log(2+sqrt(3))/8, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 06 2022
From Stefano Spezia, Mar 29 2023: (Start)
O.g.f.: 1 + (3 - 3*sqrt(1 - 4*x) - 8*x)/(2*x*sqrt(1 - 4*x)).
E.g.f.: 1 + exp(2*x)*(3*I_1(2*x) - I_0(2*x)), where I_n(x) is the modified Bessel function of the first kind.
a(n) ~ 2^(1+2*n)/sqrt(n*Pi). (End)

A215298 Number of solid standard Young tableaux of shape [[n,n-2],[2]].

Original entry on oeis.org

2, 25, 174, 962, 4804, 22689, 103510, 461318, 2021916, 8752042, 37520972, 159633060, 674969224, 2839400945, 11893509990, 49637986590, 206519808300, 856904298030, 3547095101220, 14652264350940, 60412895258040, 248675669866650, 1022088942267900, 4195255959533052

Offset: 2

Alois P. Heinz, Aug 07 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=2 of A214775.

Cf. A215002.

a:= proc(n) option remember; `if`(n=2, 2,
      2*(4*n^5 -26*n^4 +60*n^3 -58*n^2 +22*n -5)*a(n-1)/
        (2*n^5 -14*n^4 +30*n^3 -10*n^2 -31*n +25))
    end:
seq(a(n), n=2..30);

Table[12*(1 - 4*n + 10*n^2 - 8*n^3 + 2*n^4) * (2*n-4)! / ((n-2)! * (n+1)!), {n, 2, 20}] (* Vaclav Kotesovec, Sep 02 2014 *)

See Maple program.
a(n) ~ 3 * 2^(2*n-1) * sqrt(n) / sqrt(Pi). - Vaclav Kotesovec, Sep 02 2014
a(n) = 12*(1 - 4*n + 10*n^2 - 8*n^3 + 2*n^4) * (2*n-4)! / ((n-2)! * (n+1)!). - Vaclav Kotesovec, Sep 02 2014

A215299 Number of solid standard Young tableaux of shape [[n,n-3],[3]].

Original entry on oeis.org

5, 98, 962, 7020, 43573, 245962, 1305238, 6633172, 32649890, 156817044, 738717796, 3425580376, 15679951989, 70992594650, 318450985230, 1417072222020, 6261985407990, 27502477286460, 120137081521500, 522256720264680, 2260525598620770, 9746264904755652

Offset: 3

Alois P. Heinz, Aug 07 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=3 of A214775.

Cf. A215002.

a:= proc(n) option remember; `if`(n<5, [0$2, 5, 98][n],
      2*(32*n^7 -400*n^6 +1976*n^5 -4900*n^4 +6452*n^3 -4420*n^2
      +1350*n-315)*a(n-1) / (16*n^7 -224*n^6 +1204*n^5 -3008*n^4
      +2980*n^3 +1072*n^2 -4155*n +2205))
    end:
seq(a(n), n=3..30);

Flatten[{5, Table[(8*(45 - 180*n + 580*n^2 - 756*n^3 + 484*n^4 - 144*n^5 + 16*n^6) * (2*n-6)!) / (3*(n-3)!*(n+1)!), {n, 4, 20}]}] (* Vaclav Kotesovec, Sep 02 2014 *)

See Maple program.

For n > 3, a(n) = (8*(45 - 180*n + 580*n^2 - 756*n^3 + 484*n^4 - 144*n^5 + 16*n^6) * (2*n-6)!) / (3 * (n-3)! * (n+1)!). - Vaclav Kotesovec, Sep 02 2014

A215300 Number of solid standard Young tableaux of shape [[n,n-4],[4]].

Original entry on oeis.org

14, 378, 4804, 43573, 325590, 2149454, 13054108, 74688594, 408634828, 2159302420, 11097147528, 55747502501, 274790652518, 1332928973766, 6377276361900, 30149660760870, 141057202034340, 653892592144620, 3006490865152440, 13722387184879650, 62220533305358076

Offset: 4

Alois P. Heinz, Aug 07 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=4 of A214775.

Cf. A215002.

a:= proc(n) option remember; `if`(n<6, [0$3, 14, 378, 4804][n],
      ((-460961029024*n^4 +54902186125572*n^3 -347074341314956*n^2
       +421934757637074*n +6838164520124) *a(n-1) +(104238656896016*n^4
       -2317913124589048*n^3 +16535317231755832*n^2 -44274446438628908*n
       +29901662719961532)*a(n-2) +(-391233321452352*n^4
       +7447800734464704*n^3 -48294258553516272*n^2 +122447135865649584*n
       -105955729051546080)*a(n-3)) / (286655151052*n^4 -1210962058579*n^3
       +4322649356693*n^2 -24951473774234*n -30771740340558))
    end:
seq(a(n), n=4..30);

Flatten[{14,Table[(2*(n-5))!/(3*(n-5)!*(n+1)!)*(160*(-567+2394*n-8862*n^2+15592*n^3-15484*n^4+9152*n^5-3292*n^6+704*n^7-82*n^8+4*n^9)),{n,5,20}]}] (* Vaclav Kotesovec, Sep 02 2014 *)

See Maple program.

For n > 4, a(n) = (2*(n-5))!/(3*(n-5)!*(n+1)!)*(160*(-567 + 2394*n - 8862*n^2 + 15592*n^3 - 15484*n^4 + 9152*n^5 - 3292*n^6 + 704*n^7 - 82*n^8 + 4*n^9)). - Vaclav Kotesovec, Sep 02 2014

A215301 Number of solid standard Young tableaux of shape [[n,n-5],[5]].

Original entry on oeis.org

42, 1452, 22689, 245962, 2149454, 16290708, 111709178, 711996820, 4292788212, 24777783256, 138077129921, 747501664986, 3949741123174, 20444004524804, 103955714523390, 520494659493180, 2570907398453580, 12546842041060200, 60579487688891610, 289692893191143876

Offset: 5

Alois P. Heinz, Aug 07 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=5 of A214775.

Cf. A215002.

a:=proc(n) option remember; `if`(n<7, [0$4, 42, 1452, 22689][n],
   ((-940984202308081409937789248*n^7+36378423601372783158274124928*n^6
    -540987251973268278464961515672*n^5+4140452478540141056223108638628*n^4
    -17643038551017281385645661643624*n^3+40489345935054116443261823323140*n^2
    -39934974057986427003556989745680*n-247683783218781902433156798480)*a(n-1)
    +(5038765510419498883689330496*n^7-154613008671019208064714735488*n^6
    +1939670093038831522623368803072*n^5-12888788321486668402366527701360*n^4
+48941495657518683977159471709724*n^3-105016281014420890409086708155812*n^2
    +113403222542936117699329884355248*n-47046838608769352958257951122560)
    *a(n-2))/(79676793824198327746135844*n^7-1949805875384464242394656236*n^6
    +20900166698905174940775960603*n^5-125515785015357799830976856812*n^4
    +431332553464051479008795376439*n^3-723271251684163430971195319466*n^2
    +59211568171613916060478086240*n+1362260807703300463382362391640))
   end:
seq(a(n), n=5..30);

Flatten[{42, 1452, Table[(2*(n-6))! / (5 * (n-6)! * (n+1)!) * 64 * (-51975 + 217350*n - 873908*n^2 + 1738396*n^3 - 2038350*n^4 + 1500940*n^5 - 724004*n^6 + 231788*n^7 - 48750*n^8 + 6460*n^9 - 488*n^10 + 16*n^11), {n, 7, 20}]}]

See Maple program.

For n > 6, a(n) = (2*(n-6))! / (5 * (n-6)! * (n+1)!) * 64 * (-51975 + 217350*n - 873908*n^2 + 1738396*n^3 - 2038350*n^4 + 1500940*n^5 - 724004*n^6 + 231788*n^7 - 48750*n^8 + 6460*n^9 - 488*n^10 + 16*n^11). - Vaclav Kotesovec, Sep 02 2014

cp's OEIS Frontend

A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A214955 Number of solid standard Young tableaux of shape [[n,n-1],[1]].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A215298 Number of solid standard Young tableaux of shape [[n,n-2],[2]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215299 Number of solid standard Young tableaux of shape [[n,n-3],[3]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215300 Number of solid standard Young tableaux of shape [[n,n-4],[4]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215301 Number of solid standard Young tableaux of shape [[n,n-5],[5]].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula