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.

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A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 19, -39, 20, -1513, 4705, -4872, 1680, 315523, -1314807, 2052644, -1422960, 369600, -136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000, 105261234643, -661231439271, 1729495989332, -2410936679424, 1889230062720, -789044256000, 137225088000
Offset: 0

Views

Author

Peter Luschny, Sep 06 2018

Keywords

Comments

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See A318259 and the cross-references for some other members.

Examples

			[0] [         1]
[1] [        -1,         1]
[2] [        19,       -39,          20]
[3] [     -1513,      4705,       -4872,       1680]
[4] [    315523,  -1314807,     2052644,   -1422960,     369600]
[5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]

Crossrefs

Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.
Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).

Programs

  • Sage
    # uses[EW from A318259]
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))

Formula

Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1
and S(n,k) = [x^k]P(3,n), then T(n,k) = Sum_{j=0..k}((-1)^(k-j)*binomial(n-j, n-k)* Sum_{i=0..n}((-1)^i*S(n,i)*binomial(n-i,j))).

A370087 Expansion of e.g.f. exp( Sum_{k>=1} (3*k)!/k! * (x/6)^k/k ).

Original entry on oeis.org

1, 1, 11, 591, 95001, 34158801, 23091398451, 26242572454911, 46391926016144241, 120482101765570623201, 439905589366043539453851, 2180755271892747703236167151, 14267395360715456605222995351561, 120323721300147111590970495558478641
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A014606, A370085.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (3*k)!/k!*(x/6)^k/k))))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} (3*k)!/6^k * binomial(n,k) * a(n-k).

A177292 Number of permutations of 3 copies of 1..n with all adjacent differences <= 2 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 51240, 1173512, 31006978, 788946320, 18721725280, 435630242080, 9998202729808, 225310437716462
Offset: 0

Views

Author

R. H. Hardin, May 06 2010

Keywords

Comments

a(n) = (3n)!/6^n for n<=3.

Crossrefs

Cf. A014606.

Extensions

a(0) and a(11) from Alois P. Heinz, Jan 18 2016

A177293 Number of permutations of 3 copies of 1..n with all adjacent differences <= 3 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 37850400, 2626916040, 171548183912, 12520880883034, 920942694050320, 65368388837752640
Offset: 0

Views

Author

R. H. Hardin, May 06 2010

Keywords

Comments

a(n) = (3n)!/6^n for n<=4.

Crossrefs

Cf. A014606.

Extensions

a(10) from Alois P. Heinz, Jan 18 2016

A177294 Number of permutations of 3 copies of 1..n with all adjacent differences <= 4 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 41494252800, 6923871998400, 983114244363720, 139528504017728552
Offset: 0

Views

Author

R. H. Hardin, May 06 2010

Keywords

Comments

a(n) = (3n)!/6^n for n<=5.

Crossrefs

Cf. A014606.

A177296 Number of permutations of 3 copies of 1..n with all adjacent differences <= 6 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137225088000, 182509367040000, 156952841101056000, 96333779412718848000
Offset: 0

Views

Author

R. H. Hardin, May 06 2010

Keywords

Comments

a(n) = (3n)!/6^n for n<=7.

Crossrefs

Cf. A014606.

A177297 Number of permutations of 3 copies of 1..n with all adjacent differences <= 7 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137225088000, 182509367040000, 369398958888960000, 510778379991559680000
Offset: 0

Views

Author

R. H. Hardin, May 06 2010

Keywords

Comments

a(n) = (3n)!/6^n for n<=8.

Crossrefs

Cf. A014606.

A177599 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 347279, 145417911, 108103451700, 130235191847880, 237916352264650149, 626620767478218488341, 2286998940761680215812180, 11203494211631147331264432480, 71758642346800717617629677669619, 587889139521679290927189491977855371
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Cf. A014606.

Extensions

a(10) from Alois P. Heinz, Nov 01 2013
a(11)-a(13) from Alois P. Heinz, Aug 08 2018

A177600 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down.

Original entry on oeis.org

1, 1, 20, 1680, 354624, 151213748, 114062310060, 139028830444866, 256400413791435360, 680598969845995094212, 2500271405904975564019712
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Crossrefs

Cf. A014606.

Extensions

a(9)-a(10) from Alois P. Heinz, Nov 02 2013

A177606 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, down.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 167262000, 135190246573, 177591503329934, 354324970027942152
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Crossrefs

Cf. A014606.
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