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A156220 A triangle sequence of the Carlitz q-Eulerian type: p(x,n)=Product[(1 - x^k)/(1 - x), {k, 1, n}]; Q(x, n) = (-1/2)*Q(x - 1, n) + (3/2)*p(2, x - 1)^n 0
-2, -2, 3, -2, 3, -1, -2, 3, -1, 109, -2, 3, -1, 325, 1555523, -2, 3, -1, 973, 32671835, 49621794478165, -2, 3, -1, 2917, 686126051, 15630874866123949, 27744919164118690798376051, -2, 3, -1, 8749, 14408699579 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are :

{-2, 1, 0, 109, 1555848, 49621827150973, 27744919179749566350628968,

2134369240928119280766514727226937353758749,

177722377290013734874855040768840853201325647946185588862645374888, ...}.

This result is an attempt to get the Carlitz q-Eulerian recursion to work at q=2.

REFERENCES

L. Carlitz,q-Bernoulli numbers and polynomials,Duke Math. J. Volume 15, Number 4 (1948), 987-1000.http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077475200

LINKS

Table of n, a(n) for n=0..32.

FORMULA

p(x,n)=Product[(1 - x^k)/(1 - x), {k, 1, n}];

Q(x, n) = (-1/2)*Q(x - 1, n) + (3/2)*p(2, x - 1)^n

EXAMPLE

{-2},

{-2, 3},

{-2, 3, -1},

{-2, 3, -1, 109},

{-2, 3, -1, 325, 1555523},

{-2, 3, -1, 973, 32671835, 49621794478165},

{-2, 3, -1, 2917, 686126051, 15630874866123949, 27744919164118690798376051},

{-2, 3, -1, 8749, 14408699579, 4923725784550050421, 270929135785330782929292449579, 2134369240927848351630724472718209102550421},

{-2, 3, -1, 26245, 302582848643, 1550973626369423401357, 2645623010990284303964302589098643, 1313053285172771695308142008280578870609435901357, 177722377290013733561801755596066512270172647830329127278614098643},

{-2, 3, -1, 78733, 6354240293915, 488556692395327728456085, 25834508702334782928980516874334043915, 807783815771864884834631966617340166643186345540956085, 13885407572910109951736944928562883596280311211383715644970656462709043915, 126604823323645257465641899416941989179059868175498515077699244325233736491773 079148224790956085},

{-2, 3, -1, 236197, 133439047589411, 153895358106396381108660589, 252273977478303772162237881256759203839411, 496944564543772433466828016357697595649986221094241421160589, 108486363060068725671598386421907520773015062046921796271762382313441831982883 9411, 252235929900644534183482533513999780602727495055073567761673037087565728 8468629771815802085308242671160589, 612415179906389267322931264263614153788482836241860381691422341177668990205789 4666250079048081796890459409767637198354144882328839411}

MATHEMATICA

Clear[Q, e, p, n, x];

p[x_, n_] := Product[(1 - x^k)/(1 - x), {k, 1, n}]; q = 2; e[0] = 1; e[n_] := e[n] = -q*(q*e[0] + 1)^n;

Q[0, n] := -6; Q[x, 0] := 1;

Q[x_, n_] := Q[x, n] = (-1/2)*Q[x - 1, n] + (3/2)*p[2, x - 1]^n;

Clear[a]; a0 = Table[Table[ExpandAll[2^x*Q[x, n]]/3, {x, 0, m}] /. n -> m, {m, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A279632 A300817 A230140 * A261653 A083900 A113517

Adjacent sequences:  A156217 A156218 A156219 * A156221 A156222 A156223

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Feb 06 2009

STATUS

approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)