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A152534
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Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0}(1-x^k/faq(k,q)).
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2
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1, 2, 1, 3, 3, 3, 1, 5, 7, 11, 11, 8, 4, 1, 7, 13, 25, 36, 44, 42, 36, 24, 13, 5, 1, 11, 24, 54, 93, 142, 184, 215, 222, 208, 172, 126, 81, 44, 19, 6, 1, 15, 39, 98, 195, 344, 532, 753, 964, 1150, 1264, 1294, 1226, 1082, 880, 661, 451, 278, 151, 70, 26, 7, 1
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listen;
history;
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internal format)
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..63.
Eric Weisstein, q-Exponential Function from MathWorld.
Eric Weisstein, q-Factorial from MathWorld.
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FORMULA
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Sum_{k=0..binomial(n,2)}T(n,k)*q^k = Sum_{pi} faq(n,q)/Product_{i=1..n} faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1)+2*e(2)+...+n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n. Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n)) = 1.
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EXAMPLE
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1;
2,1;
3,3,3,1;
5,7,11,11,8,4,1;
7,13,25,36,44,42,36,24,13,5,1;
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MAPLE
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multinomial2q := proc(n::integer, k::integer, nparts::integer)
local lpar , res, constrp;
res := [] ;
if n< 0 or nparts <= 0 then
;
elif nparts = 1 then
if n = k then
return [[n]] ;
end if;
else
for lpar from 0 do
if lpar*nparts > n or lpar > k then
break;
end if;
for constrp in procname(n-nparts*lpar, k-lpar, nparts-1) do
if nops(constrp) > 0 then
res := [op(res), [op(constrp), lpar]] ;
end if;
end do:
end do:
end if ;
return res ;
end proc:
multinomial2 := proc(n::integer, k::integer)
local res, constrp ;
res := [] ;
for constrp in multinomial2q(n, k, n) do
if nops(constrp) > 0 then
res := [op(res), constrp] ;
end if ;
end do:
res ;
end proc:
faq := proc(i, q)
mul((q^j-1)/(q-1), j=1..i) ;
end proc;
A152534 := proc(n, k)
pi := [] ;
for sp from 0 to n do
pi := [op(pi), op(multinomial2(n, sp))] ;
end do;
tqk := 0 ;
for p in pi do
faqe :=1 ;
for i from 1 to nops(p) do
faqe := faqe* faq(i, q)^op(i, p) ;
end do:
tqk := tqk+faq(n, q)/faqe ;
end do;
tqk ;
coeftayl(tqk, q=0, k) ;
end proc:
for n from 1 to 8 do
for k from 0 to binomial(n, 2) do
printf("%d, ", A152534(n, k)) ;
end do;
printf("\n") ;
end do: # R. J. Mathar, Sep 27 2011
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CROSSREFS
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Cf. A005651(row sums), A000041(first column), A076276(second column), A152474.
Sequence in context: A099246 A039775 A178244 * A136018 A138022 A113278
Adjacent sequences: A152531 A152532 A152533 * A152535 A152536 A152537
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KEYWORD
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nonn,tabf
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AUTHOR
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Vladeta Jovovic (vladeta(AT)eunet.yu), Dec 06 2008
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STATUS
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approved
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