A069779 -id:A069779

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A069777

Array of q-factorial numbers n!_q, read by ascending antidiagonals.

+10
19

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 21, 4, 1, 1, 1, 120, 315, 52, 5, 1, 1, 1, 720, 9765, 2080, 105, 6, 1, 1, 1, 5040, 615195, 251680, 8925, 186, 7, 1, 1, 1, 40320, 78129765, 91611520, 3043425, 29016, 301, 8, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..55, flattened

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Index entries for sequences related to factorial numbers

FORMULA

T(n,q) = Product_{k=1..n} (q^k - 1) / (q - 1).

T(n,k) = Product_{n1=k..n-1} A104878(n1,k). - Johannes W. Meijer, Aug 21 2011

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, 1, ...

1, 2, 3, 4, 5, 6, 7, ...

1, 6, 21, 52, 105, 186, 301, ...

1, 24, 315, 2080, 8925, 29016, 77959, ...

1, 120, 9765, 251680, 3043425, 22661496, 121226245, ...

...

MAPLE

A069777 := proc(n, k) local n1: mul(A104878(n1, k), n1=k..n-1) end: A104878 := proc(n, k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: seq(seq(A069777(n, k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 21 2011

nmax:=9: T(0, 0):=1: for n from 1 to nmax do T(n, 0):=1: T(n, 1):= (n-1)! od: for q from 2 to nmax do for n from 0 to nmax do T(n+q, q) := product((q^k - 1)/(q - 1), k= 1..n) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od; seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Aug 21 2011

# alternative Maple program:

T:= proc(n, k) option remember; `if`(n<2, 1,

T(n-1, k)*`if`(k=1, n, (k^n-1)/(k-1)))

end:

seq(seq(T(d-k, k), k=0..d), d=0..10); # Alois P. Heinz, Sep 08 2021

MATHEMATICA

(* Returns the rectangular array *) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* Geoffrey Critzer, May 21 2017 *)

PROG

(PARI) T(n, q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018

CROSSREFS

Columns q=0..11 are A000012, A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.

Rows n=3..5 are A069778, A069779, A218503.

Main diagonal gives A347611.

Cf. A156173.

KEYWORD

easy,nonn,tabl

AUTHOR

Franklin T. Adams-Watters, Apr 07 2002

EXTENSIONS

Name edited by Michel Marcus, Sep 08 2021

STATUS

approved

A069778

q-factorial numbers 3!_q.

+10
19

1, 6, 21, 52, 105, 186, 301, 456, 657, 910, 1221, 1596, 2041, 2562, 3165, 3856, 4641, 5526, 6517, 7620, 8841, 10186, 11661, 13272, 15025, 16926, 18981, 21196, 23577, 26130, 28861, 31776, 34881, 38182, 41685, 45396, 49321, 53466, 57837, 62440, 67281, 72366

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of proper n-colorings of the 4-cycle with one vertex color fixed (offset 2). - Michael Somos, Jul 19 2002

n such that x^3 + x^2 + x + n factors over the integers. - James R. Buddenhagen, Apr 19 2005

If Y is a 4-subset of an n-set X then, for n>=5, a(n-5) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007

Equals row sums of the Connell (A001614) sequence read as a triangle. - Gary W. Adamson, Sep 01 2008

Binomial transform of 1, 5, 10, 6, 0, 0, 0 (0 continued). - Philippe Deléham, Mar 17 2014

Digital root is A251780. - Peter M. Chema, Jul 11 2016

REFERENCES

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

LINKS

Table of n, a(n) for n=0..41.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = (n + 1)*(n^2 + n + 1).

a(n) = (n+1)^3-2*T(n) where T(n) =n*(n+1)/2= A000217(n) is the n-th triangular number. - Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006

a(n) = n^8 mod (n^3+n), with offset 1..a(1)=1. - Gary Detlefs, May 02 2010

a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4), n>3. - Harvey P. Dale, Jul 11 2011

G.f.: (1+2*x+3*x^2)/(1-x)^4. - Harvey P. Dale, Jul 11 2011

For n>0 a(n) = Sum_{k=A000217(n-1)...A000217(n+1)} k. - J. M. Bergot, Feb 11 2015

E.g.f.: (1 + 5*x + 5*x^2 + x^3)*exp(x). - Ilya Gutkovskiy, Jul 11 2016

EXAMPLE

For 2-colorings only 1212 is proper so a(2-2)=1. The proper 3-colorings are: 1212,1313,1213,1312,1232,1323 so a(3-2)=6.

a(0) = 1*1 = 1;

a(1) = 1*1 + 5*1 = 6;

a(2) = 1*1 + 5*2 + 10*1 = 21;

a(3) = 1*1 + 5*3 + 10*3 + 6*1 = 52;

a(4) = 1*1 + 5*4 + 10*6 + 6*4 = 105; etc. - Philippe Deléham, Mar 17 2014

MAPLE

A069778 := proc(n)

(n+1)*(n^2+n+1) ;

end proc: # R. J. Mathar, Aug 24 2013

MATHEMATICA

LinearRecurrence[{4, -6, 4, -1}, {1, 6, 21, 52}, 41] (* or *) Table[(n + 1) (n^2 + n + 1), {n, 0, 41}] (* Harvey P. Dale, Jul 11 2011 *)

Table[QFactorial[3, n], {n, 0, 41}] (* Arkadiusz Wesolowski, Oct 31 2012 *)

PROG

(PARI) a(n)=(n+1)*(n^2+n+1)

CROSSREFS

Cf. A069777, A069779, A218503, A056108 (first differences).

Cf. A001614. - Gary W. Adamson, Sep 01 2008

Cf. A226449. - Bruno Berselli, Jun 09 2013

KEYWORD

nonn,easy

AUTHOR

Franklin T. Adams-Watters, Apr 07 2002

STATUS

approved

A218503

q-factorial numbers 5!_q.

+10
3

1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941, 37630041120, 87029433985, 188664603960, 386925380325, 756298318336, 1417430759745, 2559798038520, 4472991338725, 7589075296800, 12538953723681

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

Index entries for sequences related to factorial numbers

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

FORMULA

a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1)*(n^4 + n^3 + n^2 + n + 1).

G.f.: (1 + x*(109 + x*(8500 + x*(150700 + x*(792550 + x*(1454134 + x*(978436 + 5*x*(45788 + x*(3053 + 33*x)))))))))/(1 - x)^11.

MATHEMATICA

Table[QFactorial[5, n], {n, 0, 20}]

Join[{1}, With[{f=Times@@Table[Total[n^Range[0, i]], {i, 4}]}, Table[f, {n, 20}]]] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941}, 30] (* Harvey P. Dale, Sep 04 2017 *)

CROSSREFS

Cf. A069777, A069778, A069779.

KEYWORD

easy,nonn

AUTHOR

Arkadiusz Wesolowski, Oct 31 2012

STATUS

approved

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