A326616 -id:A326616 - cp's OEIS frontend

A326651 a(n) = Sum_{k>0} k * A326616(k,n).

0, 1, 14, 243, 9692, 865445, 196868202, 122831606807, 219073289264824, 1139077903664789577, 17597009238919048388550, 821444189426979675481201211, 116802449602563244067365434335892, 50816512870344533477388136382624158445

Offset: 0

Alois P. Heinz, Sep 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A000203, A024916, A326616, A326617.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= k-> add(n*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):
seq(a(n), n=0..15);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i*j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
a[k_] := Sum[n*Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}], {n, k, g[k]}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)

a(n) = Sum_{k=n..A024916(n)} k * A326616(k,n) = Sum_{k=n..A024916(n)} k * A326617(k,n).

A178682 The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.

Original entry on oeis.org

1, 1, 2, 5, 13, 42, 150, 576, 2266, 9966, 47466, 237019, 1224703, 6429152, 35842344, 212946552, 1325810173, 8488092454, 55276544436, 362961569008, 2465240278980, 17538501945077, 130454679958312, 1002493810175093, 7838007702606372, 61789072382062638

Offset: 0

Geoffrey Critzer, Dec 25 2010

a(n) is also the number of partitions of n where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc. There is a simple bijection between the marked partitions and the functions f. - Alois P. Heinz, Aug 30 2015

			a(3) = 5 because there are 5 such functions: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (3,3,3).
G.f. = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 42*x^5 + 150*x^6 + 576*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..680

Cf. A000110, A005651, A007837, A261774.
Main diagonal of A326500, A326616, A326617.
Row sums of A364285, A364310.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[(&+[x^(k*j)/Factorial(k*j): k in [0..m]]): j in [1..m]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jan 26 2019

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 30 2015

Range[0,20]! CoefficientList[Series[Product[Sum[x^(j i)/(j i)!,{i,0,20}],{j,1,20}],{x,0,20}],x]

m=30; my(x='x+O('x^m)); Vec(serlaplace(prod(j=1, m, sum(k=0,m, x^(k*j)/(k*j)!)))) \\ G. C. Greubel, Jan 26 2019

m = 30; T = taylor(product(sum(x^(k*j)/factorial(k*j) for k in (0..m)) for j in (1..m)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jan 26 2019

E.g.f.: Product_{j>=1} Sum_{i>=0} x^(j*i)/(j*i)!.

a(21)-a(25) from Alois P. Heinz, Aug 30 2015

A326617 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

Original entry on oeis.org

1, 1, 2, 2, 1, 5, 9, 9, 10, 9, 3, 13, 44, 96, 152, 155, 124, 140, 160, 113, 48, 16, 4, 42, 225, 680, 1350, 2180, 3751, 6050, 7420, 6870, 5555, 5330, 6300, 6475, 5025, 3000, 1250, 250, 150, 1098, 4155, 11730, 30300, 69042, 127364, 188568, 249690, 365160, 584733

Offset: 0

Alois P. Heinz, Sep 12 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...

Alois P. Heinz, Columns k = 0..40, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A178682.
Row sums give A326648.
Column sums give A326650.
Cf. A000203, A024916, A326616 (this triangle read by rows), A326649, A326651.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), n=k..g(k)), k=0..6);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

A326648 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

Original entry on oeis.org

1, 1, 2, 7, 23, 95, 481, 2515, 13130, 77546, 519770, 3641724, 25931163, 185418629, 1411248697, 11735504788, 103340890753, 931471895697, 8448978391755, 76541843977198, 715994685630321, 7110500945450780, 74757652968961770, 815423663501064107, 9012653697655462141

Offset: 0

Alois P. Heinz, Sep 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row sums of A326616 and of A326617.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n-1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n-1]], True, k++, If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k < h[n], 0, Sum[With[ {t = i j}, b[n-t, Min[n-t, i-1], k] Binomial[k, t]], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {k, h[n], n}, {i, 0, k}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

A326649 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

Original entry on oeis.org

0, 1, 4, 19, 81, 413, 2439, 14655, 86844, 573196, 4224230, 32280154, 249433713, 1925416359, 15732592327, 139542345546, 1304524118159, 12445507282579, 119198874300879, 1137647406084952, 11183828252431175, 116368970786569604, 1278400213028604214

Offset: 0

Alois P. Heinz, Sep 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A185283, A326616, A326617.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return [k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || kJean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

a(n) = Sum_{k=A185283(n)..n} k * A326616(n,k) = Sum_{k=A185283(n)..n} k * A326617(n,k).

A326650 Number of colored integer partitions using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

Original entry on oeis.org

1, 1, 5, 45, 1065, 61753, 9705069, 4394516773, 5931440509137, 24154079629381105, 300121111037478706517, 11510717148660156841731485, 1369013994385630011763634779641, 505666129597215709912984823873504809, 582167751341290615329122568805084839847101

Offset: 0

Alois P. Heinz, Sep 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Column sums of A326616 and of A326617.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= k-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):
seq(a(n), n=0..15);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k] Binomial[k, t]], {j, 0, n/i}]]];
a[k_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {n, k, g[k]}, {i, 0, k}];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A326651 a(n) = Sum_{k>0} k * A326616(k,n).

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A178682 The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.

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A326617 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

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A326648 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

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A326649 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

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A326650 Number of colored integer partitions using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

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