A152534 -id:A152534 - cp's OEIS frontend

A346980 a(n) = A152534(n,n).

1, 0, 0, 1, 8, 42, 215, 964, 4336, 18490, 78920, 327422, 1361450, 5561651, 22748172, 92064605, 372730314, 1498785098, 6028277742, 24137078144, 96650940881, 385889337069, 1540641044744, 6139148609668, 24462365837156, 97350374411004, 387401219371335

Offset: 0

Alois P. Heinz, Aug 09 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Main diagonal of A152534.

Cf. A346981.

f:= proc(n) option remember; `if`(n<2, 1, f(n-1)*(q^n-1)/(q-1)) end:
b:= proc(n, i) option remember; simplify(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)/f(i)^j, j=0..n/i)))
    end:
a:= n-> coeff(simplify(f(n)*b(n$2)), q, n):
seq(a(n), n=0..26);

f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]/f[i]^j, {j, 0, n/i}]];
a[n_] := SeriesCoefficient[f[n]*b[n, n], {q, 0, n}];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Apr 07 2022, after Alois P. Heinz *)

a(n) = A152534(n,n).

A152536 a(n) = Sum_{k=0..binomial(n,2)} (-1)^k*A152534(n,k).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 47, 110, 246, 617, 1602, 4035, 11481, 29927, 95503, 249926, 871030, 2316317, 8879558, 23674841, 98329551, 264329177, 1191578522, 3207278255, 15543026747, 42011308653, 218668538441, 591460307157, 3285749117475, 8905905152798, 52700813279423

Offset: 0

Vladeta Jovovic, Dec 06 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A152534.

A152536 := proc(n)
    add( (-1)^k*A152534(n,k),k=0..binomial(n,2)) ;
end proc: # R. J. Mathar, Jun 15 2013

a(17)-a(20) from Vaclav Kotesovec, May 24 2020

a(0), a(21)-a(32) from Alois P. Heinz, Aug 09 2021

A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)q^k = n!Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

Original entry on oeis.org

1, 1, 3, 1, 13, 8, 8, 1, 73, 63, 89, 78, 41, 15, 1, 501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1, 4051, 5225, 9734, 13799, 18709, 20441, 20520, 18101, 14831, 10200, 5891, 3199, 1109, 314, 35, 1, 37633, 55656, 112370, 177457, 270746, 352969, 442897

Offset: 0

Vladeta Jovovic, Dec 05 2008

			Triangle T(n,k) begins:
    1;
    1;
    3,   1;
   13,   8,   8,    1;
   73,  63,  89,   78,   41,  15,   1;
  501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1;
  ...

Alois P. Heinz, Rows n = 0..36, flattened

Cf. A000262 (first column), A105219(second column), A137341 (row sums), A152534.

T(n,n) gives A346981.

{T(n,k)=local(e_q=sum(j=0,n,x^j/prod(i=1,j,(q^i-1)/(q-1)))+x*O(x^n)); n!*polcoeff(polcoeff(exp(e_q-1),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)} \\ Paul D. Hanna, Dec 15 2008

Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = n!. - Vladeta Jovovic, Dec 05 2008
From Paul D. Hanna, Dec 15 2008: (Start)
E.g.f.: A(x,q) = exp(e_q(x,q) - 1) = Sum_{n>=0} Sum_{k=0..n(n-1)/2} T(n,k)*q^k*x^n/(n!*faq(n,q)) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) and faq(n,q) = Product_{j=1..n} (q^j-1)/(q-1) with faq(0,q)=1.
Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = n!*A000110((n+1)/2), where A000110 is the Bell numbers. (End)

T(0,0)=1 prepended by Alois P. Heinz, Feb 04 2018

cp's OEIS Frontend

A346980 a(n) = A152534(n,n).

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A152536 a(n) = Sum_{k=0..binomial(n,2)} (-1)^k*A152534(n,k).

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A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)q^k = n!Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

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cp's OEIS Frontend

A346980 a(n) = A152534(n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A152536 a(n) = Sum_{k=0..binomial(n,2)} (-1)^k*A152534(n,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Extensions

A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)*q^k = n!*Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!*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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)q^k = n!Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.