A090802 -id:A090802 - cp's OEIS frontend

A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.

1, 7, 1, 49, 14, 2, 343, 147, 42, 6, 2401, 1372, 588, 168, 24, 16807, 12005, 6860, 2940, 840, 120, 117649, 100842, 72030, 41160, 17640, 5040, 720, 823543, 823543, 705894, 504210, 288120, 123480, 35280, 5040, 5764801, 6588344, 6588344, 5647152, 4033680, 2304960, 987840, 282240, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=7. Also:
first column:    A000420;
second column:   A027473;
third column:  2*A027474;
fourth column: 6*A140107.

			Triangle begins:
1;
7,       1;
49,      14,      2;
343,     147,     42,      6;
2401,    1372,    588,     168,     24;
16807,   12005,   6860,    2940,    840,     120;
117649,  100842,  72030,   41160,   17640,   5040,    720;
823543,  823543,  705894,  504210,  288120,  123480,  35280,  5040; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000420, A027466, A027473, A027474, A090802, A140107, A217629, A218016.

[Factorial(n)/Factorial(n-k)*7^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*7^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 7^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(7x)*x^k.

A052771 E.g.f.: x^3*exp(x)^2.

Original entry on oeis.org

0, 0, 0, 6, 48, 240, 960, 3360, 10752, 32256, 92160, 253440, 675840, 1757184, 4472832, 11182080, 27525120, 66846720, 160432128, 381026304, 896532480, 2091909120, 4844421120, 11142168576, 25467813888, 57881395200, 130862284800, 294440140800, 659545915392

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The old definition of this sequence was "A simple grammar".

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 728.

Cf. A090802.

[n*(n-1)*(n-2)/8*2^n: n in [0..30]]; // Vincenzo Librandi, Dec 06 2012

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Range[0, 30]! CoefficientList[Series[Exp[x]^2 x^3, {x, 0, 30}], x] (* Vincenzo Librandi, Dec 06 2012 *)

a(n) = A090802(n, 3).
Recurrence: {a(1)=0, a(2)=0, a(3)=6, (-2*n-2)*a(n)+(-2+n)*a(n+1)}.
a(n) = n*(n-1)*(n-2)/8 * 2^n. - Vaclav Kotesovec, Nov 27 2012
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4). - Chai Wah Wu, May 25 2016
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=3} 1/a(n) = log(2) - 1/2.
Sum_{n>=3} (-1)^(n+1)/a(n) = 9*log(3/2) - 7/2. (End)

New definition by Bruno Berselli, Dec 06 2012

More terms from Vincenzo Librandi, Dec 06 2012

A052796 Expansion of e.g.f. x^4*exp(x)^2.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 1440, 6720, 26880, 96768, 322560, 1013760, 3041280, 8785920, 24600576, 67092480, 178913280, 467927040, 1203240960, 3048210432, 7620526080, 18827182080, 46022000640, 111421685760, 267412045824, 636695347200, 1504916275200

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) is the number of ways that n people can form two distinct committees and then choose a president and vice president for each committee.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 753.

Cf. A090802.

[(n-3)*(n-2)*(n-1)*n * 2^(n-4): n in [0..30]]; // Vincenzo Librandi, Dec 06 2012

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Range[0, 30]!* CoefficientList[Series[Exp[x]^2 * x^4,{x, 0, 30}], x] (* Vincenzo Librandi, Dec 06 2012 *)

E.g.f.: x^4*exp(x)^2.
a(n) = A090802(n, 4).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-2*n-2)*a(n)+(n-3)*a(n+1)}.
O.g.f.: -24*x^4/(2*x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
a(n) = (n-3)*(n-2)*(n-1)*n * 2^(n-4). - Vaclav Kotesovec, Nov 27 2012
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=4} 1/a(n) = 5/18 - log(2)/3.
Sum_{n>=4} (-1)^n/a(n) = 9*log(3/2) - 65/18. (End)

More terms from Vincenzo Librandi, Dec 06 2012

A082569 a(1)=2; a(n)=ceiling(n*(a(n-1)-1/a(n-1))).

Original entry on oeis.org

2, 3, 8, 32, 160, 960, 6720, 53760, 483840, 4838400, 53222400, 638668800, 8302694400, 116237721600, 1743565824000, 27897053184000, 474249904128000, 8536498274304000, 162193467211776000, 3243869344235520000

Offset: 1

Benoit Cloitre, May 06 2003

nonn

Equals 4 * A002301.

nxt[{n_,a_}]:={n+1,Ceiling[(n+1)(a-1/a)]}; Transpose[NestList[nxt,{1,2},20]][[2]] (* Harvey P. Dale, May 30 2015 *)

a(1)=2 a(2)=3 and for n>2, a(n)= 4*n!/3.

a(n) = A090802(n, n-3) for n > 2. - Ross La Haye, Sep 26 2005

A189071 The n-th derivative of x^10 evaluated at x=2.

Original entry on oeis.org

1024, 5120, 23040, 92160, 322560, 967680, 2419200, 4838400, 7257600, 7257600, 3628800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Vincenzo Librandi, Apr 21 2011

This is the 10th row of A090802.

Index entries for linear recurrences with constant coefficients, signature (1).

[n lt 1 select 1024 else n lt 11 select (1/2)*(11-n)*Self(n) else 0: n in [0..67]];  // Bruno Berselli, Sep 08 2011

LinearRecurrence[{1},{1024, 5120, 23040, 92160, 322560, 967680, 2419200, 4838400, 7257600, 7257600, 3628800, 0},68] (* Ray Chandler, Jul 15 2015 *)

a(n)=if(n<11,10!/(10-n)!*2^(10-n)) \\ Charles R Greathouse IV, Sep 08 2011

a(n) = (1/2)*(11-n)*a(n-1), a(0)=1024. - Bruno Berselli, Sep 08 2011

A359110 Number of Boolean monoids of order 2^n up to isomorphism.

Original entry on oeis.org

1, 5, 83, 242547

Offset: 1

Choiwah Chow, Dec 18 2022

MathStructures, Boolean Monoids.

Cf. A090802, A066534, A198085, A342908, A058129, A058129.

cp's OEIS Frontend

A218017 Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.

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A052771 E.g.f.: x^3*exp(x)^2.

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Maple

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A052796 Expansion of e.g.f. x^4*exp(x)^2.

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Magma

Maple

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A082569 a(1)=2; a(n)=ceiling(n*(a(n-1)-1/a(n-1))).

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Mathematica

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A189071 The n-th derivative of x^10 evaluated at x=2.

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Magma

Mathematica

PARI

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A359110 Number of Boolean monoids of order 2^n up to isomorphism.

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A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.