cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A333446 Table T(n,k) read by upward antidiagonals. T(n,k) = Sum_{i=1..n} Product_{j=1..k} (i-1)*k+j.

Original entry on oeis.org

1, 3, 2, 6, 14, 6, 10, 44, 126, 24, 15, 100, 630, 1704, 120, 21, 190, 1950, 13584, 30360, 720, 28, 322, 4680, 57264, 390720, 666000, 5040, 36, 504, 9576, 173544, 2251200, 14032080, 17302320, 40320, 45, 744, 17556, 428568, 8626800, 110941200, 603353520, 518958720, 362880
Offset: 1

Views

Author

Chai Wah Wu, Mar 23 2020

Keywords

Comments

T(n,k) is the maximum value of Sum_{i=1..n} Product_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}. For the minimum value see A331889.

Examples

			From _Seiichi Manyama_, Jul 23 2020: (Start)
T(3,2) = Sum_{i=1..3} Product_{j=1..2} (i-1)*2+j = 1*2 + 3*4 + 5*6 = 44.
Square array begins:
   1,   2,    6,     24,      120,        720, ...
   3,  14,  126,   1704,    30360,     666000, ...
   6,  44,  630,  13584,   390720,   14032080, ...
  10, 100, 1950,  57264,  2251200,  110941200, ...
  15, 190, 4680, 173544,  8626800,  538459200, ...
  21, 322, 9576, 428568, 25727520, 1940869440, ... (End)

Links

Crossrefs

Column k=1-3 give A000217, A268684, A268685(n-1).
Main diagonal gives A336513.
Cf. A323663, A331889,

Programs

  • Python
    def T(n,k): # T(n,k) for A333446
    c, l = 0, list(range(1,k*n+1,k))
    lt = list(l)
    for i in range(n):
        for j in range(1,k):
            lt[i] *= l[i]+j
        c += lt[i]
    return c

Formula

T(n,k) = Sum_{i=1..n} Gamma(ik+1)/Gamma((i-1)k+1).

A336502 Partial sums of A057003.

Original entry on oeis.org

1, 7, 127, 5167, 365527, 39435607, 6006997207, 1226103906007, 322796982334807, 106460296033918807, 42980408446129381207, 20846482682939051365207, 11959807608801430284133207, 8010447502346968140207973207, 6193994326661240674349352805207, 5476021766725276671842502543205207
Offset: 1

Views

Author

Seiichi Manyama, Jul 23 2020

Keywords

Comments

Inspired by doubly triangular numbers (A002817).

Examples

			a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3 + 4*5*6 = 127.
a(4) = 1 + 2*3 + 4*5*6 + 7*8*9*10 = 5167.

Links

Crossrefs

Cf. A000217, A002817, A007489, A057003, A227364, A336513.

Programs

  • Mathematica
    Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* Amiram Eldar, Jul 23 2020 *)
  • PARI
    {a(n) = sum(i=1, n, prod(j=(i-1)*i/2+1, i*(i+1)/2, j))}

Formula

a(n) = Sum_{i=1..n} Product_{j=T(i-1)+1..T(i)} j where T(n) is n-th triangular number.
a(n) = A227364(T(n)) where T(n) is n-th triangular number.
a(n) ~ n^(2*n) / 2^n. - Vaclav Kotesovec, Nov 20 2021

A349480 a(n) = Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.

Original entry on oeis.org

1, 1, 10, 390, 33456, 4845360, 1059099840, 325460948400, 133697543616000, 70733019878196480, 46831083260349024000, 37927830201482962540800, 36883442511877368877747200, 42409212946187708288828160000
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2021

Keywords

Examples

			a(2) = -1*2 + 3*4 = 10.
a(3) = 1*2*3 - 4*5*6 + 7*8*9 = 390.
a(4) = -1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16 = 33456.

Links

Crossrefs

Cf. A336513, A349470.

Programs

  • Mathematica
    a[n_] := n! * Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k));

Formula

a(n) = n! * A349470(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).
Showing 1-3 of 3 results.

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