A204856 -id:A204856 - cp's OEIS frontend

A367955 Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1

Offset: 0

Alois P. Heinz, Dec 05 2023

Rows and also columns reversed converge to A365441.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.

			T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  .  1;
  .  .  1, 1;
  .  .  .  1, 1, 2, 1;
  .  .  .  .  1, 1, 2, 5, 2,  3,  1;
  .  .  .  .  .  1, 1, 2, 5, 10,  7,  7, 11,  3,  4,  1;
  .  .  .  .  .  .  1, 1, 2,  5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Partition of a set

Row sums give A000110.
Column sums give A204856.
Antidiagonal sums give A368102.
T(2n,3n) gives A365441.
T(n,2n) gives A368675.
Row maxima give A367969.
Row n has A000124(n-1) terms (for n>=1).
Cf. A000217, A124327 (the same for block minima), A200660, A278677.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
T[0, 0] = 1; T[n_, k_] := b[k, n, 0];
Table[Table[T[n, k], {k, n, n*(n + 1)/2}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz's second Maple program *)

Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n-1) for n>=1.
Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
T(n,n) = T(n,n*(n+1)/2) = 1.

A291960 Triangle read by rows: T(n,k) = T(n-k,k-1) + k * T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 3, 1, 0, 1, 7, 1, 0, 1, 7, 3, 0, 1, 15, 6, 0, 1, 15, 10, 1, 0, 1, 31, 16, 1, 0, 1, 31, 33, 3, 0, 1, 63, 45, 6, 0, 1, 63, 79, 14, 0, 1, 127, 130, 20, 1, 0, 1, 127, 198, 45, 1, 0, 1, 255, 300, 69, 3, 0, 1, 255, 517, 135

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1,  1;
  0, 1,  1;
  0, 1,  3;
  0, 1,  3,  1;
  0, 1,  7,  1;
  0, 1,  7,  3;
  0, 1, 15,  6;
  0, 1, 15, 10, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A204856.
Columns 0-2 give A000007, A000012, A052551(n-3).
Cf. A008289, A216652, A291969.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-j*x^j).

A204858 G.f.: Sum_{n>=0} n! * x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 13, 21, 33, 67, 115, 183, 333, 541, 937, 1635, 2643, 4327, 7573, 12069, 20025, 33427, 54259, 87375, 144669, 231541, 374809, 607443, 970539, 1545367, 2502205, 3947541, 6270057, 9997867, 15776083, 24832503, 39351309, 61552501, 96632689

Offset: 0

Paul D. Hanna, Jan 20 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 +...
where A(x) = 1 + x/(1-x) + 2!*x^3/((1-x)*(1-2*x^2)) + 3!*x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + 4!*x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..6263

Cf. A032020, A204857, A204856.

Table[SeriesCoefficient[Sum[n!*x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {n, 0, 100}], {x, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 19 2017 *)

{a(n)=polcoeff(1+sum(m=1,n,m!*x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}

G.f.: 1/(1 - x/(1 - 2*x^2*(1-x)/(1 - 3*x^3*(1-2*x^2)/(1 - 4*x^4*(1-3*x^3)/(1 - 5*x^5*(1-4*x^4)/(1 - 6*x^6*(1-5*x^5)/(1 -...))))))), a continued fraction.
From Vaclav Kotesovec, Jun 18 2019: (Start)
a(n) ~ c * 3^(n/3), where
c = 8007.60951343849770902289074154120578227939552369... if mod(n,3)=0
c = 8007.30566699919825273673656299755925992856381905... if mod(n,3)=1
c = 8007.19663204881021378993302255541874790731157021... if mod(n,3)=2
(End)

A204855 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)x^k).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 54, 120, 278, 666, 1671, 4355, 11804, 33019, 94960, 279219, 836907, 2550991, 7901818, 24875931, 79667065, 259892494, 864832484, 2938862050, 10204420451, 36199678110, 131086662067, 483853193560, 1817012289562, 6927980565530

Offset: 0

Paul D. Hanna, Jan 20 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 25*x^7 +...
where A(x) = 1 + x/(1-x) + x^3/((1-2*x)*(1-x^2)) + x^6/((1-3*x)*(1-2*x^2)*(1-x^3)) + x^10/((1-4*x)*(1-3*x^2)*(1-2*x^3)*(1-x^4)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A204856, A204857.

{a(n)=polcoeff(1+sum(m=1,n,x^(m*(m+1)/2)/prod(k=1,m,1-(m-k+1)*x^k+x*O(x^n))),n)}

A318770 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - j*x^j).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 8, 9, 17, 19, 38, 42, 80, 97, 174, 208, 389, 460, 826, 1049, 1790, 2248, 3989, 4933, 8451, 11116, 18300, 23742, 40446, 51774, 85774, 115454, 184806, 245967, 406768, 533210, 860295, 1179570, 1850325, 2505585, 4046594, 5407269, 8556317, 11877833, 18327723

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

Cf. A193196, A204856, A318771.

a:=series(add(x^(k^2)/mul((1-j*x^j),j=1..k),k=0..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019

nmax = 46; CoefficientList[Series[Sum[x^k^2/Product[(1 - j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306704 Expansion of Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 + jx^j).

Original entry on oeis.org

1, 1, -1, 2, -2, 0, 1, 3, -5, -6, 11, 11, -12, -35, 33, 35, -22, -102, 170, 47, -224, -491, 874, 695, -598, -2606, 2246, 1503, -664, -6420, 11590, 2526, -13762, -34647, 61785, 37119, -32372, -181052, 147105, 104896, 12824, -436333, 799007, -109587, -868230, -2316921, 4447531

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A003406, A204856.

nmax = 46; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306663 Expansion of Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - jx^j).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 17, 17, 35, 38, 74, 80, 161, 173, 336, 387, 713, 818, 1555, 1765, 3248, 3923, 6905, 8282, 15012, 17814, 31419, 39321, 66679, 82923, 144789, 177721, 302789, 390123, 642640, 821316, 1390825, 1755400, 2910638, 3833338, 6165743, 8060128, 13322378

Offset: 0

Ilya Gutkovskiy, Mar 04 2019

nonn

Cf. A003106, A193196, A204856.

nmax = 48; CoefficientList[Series[Sum[x^(k (k + 1))/Product[(1 - j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306732 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 5, 4, 7, 8, 5, 7, 11, 15, 22, 17, 31, 39, 20, 31, 39, 64, 81, 85, 125, 97, 170, 211, 121, 167, 229, 265, 385, 531, 548, 573, 814, 686, 1150, 1339, 860, 1131, 1344, 1888, 2109, 2780, 3656, 4127, 4294, 4498, 6320, 5568, 8747, 10260, 6856, 8673, 10580

Offset: 0

Ilya Gutkovskiy, Mar 06 2019

nonn

Cf. A053261, A204856, A306704.

nmax = 59; CoefficientList[Series[Sum[x^(k (k + 1)/2) Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A367955 Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A291960 Triangle read by rows: T(n,k) = T(n-k,k-1) + k * T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A204858 G.f.: Sum_{n>=0} n! * x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A204855 G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)*x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A318770 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - j*x^j).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A306704 Expansion of Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 + j*x^j).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A306663 Expansion of Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - j*x^j).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A306732 Expansion of Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + j*x^j).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A204858 G.f.: Sum_{n>=0} n! * x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

A204855 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)x^k).

A306704 Expansion of Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 + jx^j).

A306663 Expansion of Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - jx^j).

A306732 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + j*x^j).