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 11-18 of 18 results.

A215549 Number of standard Young tableaux of shape [9n,9].

Original entry on oeis.org

0, 4862, 2466750, 63882940, 670609940, 4277470470, 19794795118, 73143988775, 228723580800, 628737007195, 1559830082888, 3559370252529, 7576971259000, 15210525840125, 29040055455840, 53087119860346, 93432350566520, 159028880903100, 262755041438890
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Comments

Also the number of binary words with 9n 1's and 9 0's such that for every prefix the number of 1's is >= the number of 0's.

Links

Crossrefs

Row n=9 of A214776.

Programs

  • Maple
    a:= n-> max(0, binomial(9*n+9,9)*(9*n-8)/(9*n+1)):
seq(a(n), n=0..30);

Formula

G.f.: (8*x^9 -80*x^8 +37540*x^7 +3833365*x^6 +48377194*x^5 +151114390*x^4 +142200850*x^3 +39434230*x^2 +2418130*x +4862)*x / (x-1)^10.
a(n) = C(9*n+9,9)*(9*n-8)/(9*n+1) for n>0, a(0) = 0.

A215550 Number of standard Young tableaux of shape [10n,10].

Original entry on oeis.org

0, 16796, 15737865, 574221648, 7766844470, 60610884906, 331670995656, 1414591812920, 5014392953273, 15408077648040, 42254306265171, 105611585968616, 244384627765200, 529868222188998, 1086607184873210, 2123449623259536, 3978448975695051, 7182177974166580
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Comments

Also the number of binary words with 10n 1's and 10 0's such that for every prefix the number of 1's is >= the number of 0's.

Links

Crossrefs

Row n=10 of A214776.

Programs

  • Maple
    a:= n-> max(0, binomial(10*n+10,10)*(10*n-9)/(10*n+1)):
seq(a(n), n=0..30);

Formula

G.f.: (9*x^10 -99*x^9 -140503*x^8 -25387417*x^7 -510202946*x^6 -2566871318*x^5 -4166581331*x^4 -2313217577*x^3 -402028913*x^2 -15553109*x -16796)*x / (x-1)^11.
a(n) = C(10*n+10,10)*(10*n-9)/(10*n+1) for n>0, a(0) = 0.

A215551 a(n) = binomial(7*n,n)*(5*n+1)/(6*n+1).

Original entry on oeis.org

1, 6, 77, 1120, 17199, 272272, 4395118, 71916768, 1188576675, 19794795118, 331670995656, 5584693695280, 94418718611490, 1601780734932840, 27253077978559384, 464859662065426880, 7946606112855555915, 136106890528701061242, 2335187812028912966125
Offset: 0

Views

Author

Alois P. Heinz, Aug 15 2012

Keywords

Comments

Number of standard Young tableaux of shape [6n,n].

Links

Crossrefs

Column k=6 of A214776.

Programs

  • Maple
    a:= n-> binomial(7*n,n)*(5*n+1)/(6*n+1):
seq(a(n), n=0..20);
  • Mathematica
    Table[Binomial[7n,n] (5n+1)/(6n+1),{n,0,30}] (* Harvey P. Dale, Apr 30 2022 *)

Formula

a(n) = C(7*n,n)*(5*n+1)/(6*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(5*n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ sqrt(3) * 5 * 7^(7*n+1/2) / (6^(6*n+2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215552 a(n) = binomial(8*n,n)*(6*n+1)/(7*n+1).

Original entry on oeis.org

1, 7, 104, 1748, 31000, 566618, 10559208, 199448964, 3804949176, 73143988775, 1414591812920, 27492340515912, 536480138597624, 10504551860174120, 206284010045343000, 4061109502392133464, 80126310234711780600, 1583953257985260802200, 31365436013686385802048
Offset: 0

Views

Author

Alois P. Heinz, Aug 15 2012

Keywords

Comments

Number of standard Young tableaux of shape [7n,n].

Links

Crossrefs

Column k=7 of A214776.

Programs

  • Maple
    a:= n-> binomial(8*n,n)*(6*n+1)/(7*n+1):
seq(a(n), n=0..20);
  • Mathematica
    Table[Binomial[8n,n] (6n+1)/(7n+1),{n,0,20}] (* Harvey P. Dale, Mar 30 2014 *)

Formula

a(n) = C(8*n,n)*(6*n+1)/(7*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(6*n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ 3 * 4^(12*n+1) / (7^(7*n+3/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215553 a(n) = binomial(9*n,n)*(7*n+1)/(8*n+1).

Original entry on oeis.org

1, 8, 135, 2574, 51765, 1072764, 22664655, 485325150, 10495906641, 228723580800, 5014392953273, 110471247622008, 2443644577217025, 54238301397083592, 1207358254510786125, 26943600312354592800, 602594302548520646793, 13502992968597378745800
Offset: 0

Views

Author

Alois P. Heinz, Aug 15 2012

Keywords

Comments

Number of standard Young tableaux of shape [8n,n].

Links

Crossrefs

Column k=8 of A214776.

Programs

  • Maple
    a:= n-> binomial(9*n,n)*(7*n+1)/(8*n+1):
seq(a(n), n=0..20);
  • Mathematica
    a[n_] := Binomial[9*n,n]*(7*n+1)/(8*n+1); Array[a, 21, 0] (* Amiram Eldar, Aug 29 2025 *)

Formula

a(n) = C(9*n,n)*(7*n+1)/(8*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(7*n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ 7 * 3^(18*n+1) / (2^(24*n+5) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215554 a(n) = binomial(10*n,n)*(8*n+1)/(9*n+1).

Original entry on oeis.org

1, 9, 170, 3625, 81510, 1888460, 44602348, 1067658735, 25810820750, 628737007195, 15408077648040, 379444514503119, 9382177773301060, 232775087755980000, 5792018711632340160, 144481310070897555910, 3611955405113118024990, 90470699668284950782170
Offset: 0

Views

Author

Alois P. Heinz, Aug 15 2012

Keywords

Comments

Number of standard Young tableaux of shape [9n,n].

Links

Crossrefs

Column k=9 of A214776.

Programs

  • Maple
    a:= n-> binomial(10*n,n)*(8*n+1)/(9*n+1):
seq(a(n), n=0..20);
  • Mathematica
    Table[Binomial[10n,n] (8n+1)/(9n+1),{n,0,20}] (* Harvey P. Dale, Dec 19 2016 *)

Formula

a(n) = C(10*n,n)*(8*n+1)/(9*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(8*n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ 2^(10*n+3) * 5^(10*n+1/2) / (3^(18*n+3) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215555 a(n) = binomial(11*n,n)*(9*n+1)/(10*n+1).

Original entry on oeis.org

1, 10, 209, 4928, 122507, 3137706, 81921840, 2167714560, 57928578191, 1559830082888, 42254306265171, 1150225193717600, 31437550449182800, 862165662720962754, 23713320186494219820, 653855026849948319616, 18068367354658442882775, 500254126810079793897130
Offset: 0

Views

Author

Alois P. Heinz, Aug 15 2012

Keywords

Comments

Number of standard Young tableaux of shape [10n,n].

Links

Crossrefs

Column k=10 of A214776.

Programs

  • Maple
    a:= n-> binomial(11*n,n)*(9*n+1)/(10*n+1):
seq(a(n), n=0..20);
  • Mathematica
    Table[Binomial[11n,n] (9n+1)/(10n+1),{n,0,20}] (* Harvey P. Dale, Nov 21 2023 *)

Formula

a(n) = C(11*n,n)*(9*n+1)/(10*n+1).
a(n) ~ 9 * 11^(11*n+1/2) / (4^(5*n+1) * 5^(10*n+3/2) * (sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215557 Number of standard Young tableaux of shape [n^2,n].

Original entry on oeis.org

1, 1, 9, 154, 3705, 115101, 4395118, 199448964, 10495906641, 628737007195, 42254306265171, 3148956023335200, 257758558133120135, 22991045919047089170, 2219652431230209792300, 230617851021799852486856, 25657807699789594931790369, 3043509929953923167586547335
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Comments

Also the number of binary words with n^2 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.

Links

Crossrefs

Main diagonal of A214776.

Programs

  • Maple
    a:= n-> binomial((n+1)*n, n)*((n-1)*n+1)/(n*n+1):
seq(a(n), n=0..20);
  • Mathematica
    Table[Binomial[n(n+1),n] (n(n-1)+1)/(n^2+1),{n,0,20}] (* Harvey P. Dale, Dec 08 2023 *)

Formula

a(n) = C((n+1)*n, n)*((n-1)*n+1)/(n*n+1).
a(n) = A214776(n,n).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n^2-n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ n^(n-1/2) * exp(n+1/2-1/(6*n)) / sqrt(2*Pi). - Amiram Eldar, Aug 29 2025
Previous Showing 11-18 of 18 results.

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

Content is available under The OEIS End-User License Agreement.