A091360 -id:A091360 - cp's OEIS frontend

A340793 Sequence whose partial sums give A000203.

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 7, 9, 5, 2, 1, 11, 18, 11, 5, 2, 1, 15, 30, 22, 11, 5, 2, 1, 22, 53, 42, 24, 11, 5, 2, 1, 30, 85, 78, 46, 24, 11, 5, 2, 1, 42, 139, 138, 90, 48, 24, 11, 5, 2, 1, 56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1, 77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1, 101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1

Offset: 1

Christian G. Bower, Jan 02 2004

Row sums give A000219.
Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.
Columns converge to A091360.

			Triangle starts:
01:  1,
02:  2, 1,
03:  3, 2, 1,
04:  5, 5, 2, 1,
05:  7, 9, 5, 2, 1,
06:  11, 18, 11, 5, 2, 1,
07:  15, 30, 22, 11, 5, 2, 1,
08:  22, 53, 42, 24, 11, 5, 2, 1,
09:  30, 85, 78, 46, 24, 11, 5, 2, 1,
10:  42, 139, 138, 90, 48, 24, 11, 5, 2, 1,
11:  56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1,
12:  77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1,
13:  101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1,
14:  135, 760, 1088, 899, 580, 336, 182, 96, 48, 24, 11, 5, 2, 1,
15:  176, 1115, 1741, 1512, 1020, 606, 340, 182, 96, 48, 24, 11, 5, 2, 1,
...

Alois P. Heinz, Rows n = 1..141, flattened

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      min(d, k)*d, d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 15 2014

(* load EulerTransform from 'seqtranslib.m' under OEIS-Transforms *) Table[EulerTransform[Table[Min[c, r], {r, 20}]] - EulerTransform[Table[Min[c-1, r], {r, 20}]], {c, 20}] // Transpose
(* second program: *)
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k]*d, {d, Divisors[j]}] *A[n-j, k], {j, 1, n}]/n]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1] ]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

k-th column is EulerTransform[1, 2, 3, .., k, k, k, ..]-EulerTransform[1, 2, 3, .., k-1, k-1, k-1, ..]. - Wouter Meeussen, Aug 29 2004

Definition corrected, Joerg Arndt, Jul 21 2014

A091356 Number of planar partitions of n with exactly 2 rows.

Original entry on oeis.org

1, 2, 5, 9, 18, 30, 53, 85, 139, 215, 336, 504, 760, 1115, 1635, 2351, 3375, 4770, 6725, 9368, 13006, 17885, 24510, 33319, 45139, 60743, 81457, 108610, 144334, 190844, 251542, 330082, 431825, 562710, 731154, 946644, 1222305, 1573155, 2019471

Offset: 2

Christian G. Bower, Jan 02 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column 2 of A091355.

Cf. A000219, A091357, A091358, A091359, A091360.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
     *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, 2)-b(n, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Oct 02 2018

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k] d, {d, Divisors[j]}] b[n - j, k], {j, 1, n}]/n];
a[n_] :=  b[n, 2] - b[n, 1];
a /@ Range[2, 50] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

a(n) = A000990(n) - A000041(n).

A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

Original entry on oeis.org

1, 4, 1, 11, 2, 1, 28, 7, 2, 1, 62, 15, 5, 2, 1, 137, 38, 13, 5, 2, 1, 278, 76, 28, 11, 5, 2, 1, 561, 164, 60, 26, 11, 5, 2, 1, 1080, 316, 124, 52, 24, 11, 5, 2, 1, 2051, 623, 244, 108, 50, 24, 11, 5, 2, 1, 3778, 1156, 469, 208, 100, 48, 24, 11, 5, 2, 1, 6885, 2160, 886, 404, 194, 98, 48, 24, 11, 5, 2, 1

Offset: 1

Wouter Meeussen, Feb 01 2004

For large n the rows end in A091360 = partial sums of A000219 (count of plane partitions).

			Triangle begins:
    1;
    4,  1;
   11,  2,  1;
   28,  7,  2,  1;
   62, 15,  5,  2,  1;
  137, 38, 13,  5,  2,  1;
  ...

Alois P. Heinz, Rows n = 1..50, flattened
M. F. Hasler, A092288, rows 1 - 18.

Column k=1 gives A090539.
Row sums give A319648.
T(2n+1,n+1) gives A091360.
Cf. A000219, A066633.

Table[Length /@ Split[Sort[Flatten[planepartitions[k]]]], {k, 12}]

A092288_row(n, c=vector(n), m, k)={for(i=1, #n=PlanePartitions(n), for(j=1,#m=n[i], for(i=1,#k=m[j], c[k[i]]++))); c} \\ See A091298 for PlanePartitions(). See below for more efficient code.
M92288=[]; A092288(n,k,L=0)={n>1||return(if(L,[n,n==k],n==k)); if(#L&& #L<3, my(j=setsearch(M92288,[[n,k,L],[]],1)); j<=#M92288&& M92288[j][1]==[n,k,L]&& return(M92288[j][2])); my(c(p)=sum(i=1,#p,p[i]==k),S=[0,0],t); for(m=1,n,my(P=if(L,select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(m,L[1],#L)), partitions(m))); if(mA092288(n-m,k,Vecrev(P[i])); S+=[t[1], t[1]*c(P[i])+t[2]], S+=[#P,vecsum(apply(c,P))])); if(L, #L<3&& M92288= setunion(M92288,[[[n,k,L],S]]);S,S[2])} \\ M. F. Hasler, Sep 26 2018

A302830 Expansion of (1/(1 - x))Product_{k>=1} 1/(1 - kx^k).

Original entry on oeis.org

1, 2, 5, 11, 25, 50, 106, 203, 401, 755, 1427, 2597, 4804, 8566, 15352, 27027, 47551, 82187, 142445, 243025, 414919, 700739, 1181236, 1972552, 3293898, 5450728, 9008081, 14791741, 24244399, 39494615, 64266141, 103979929, 167991853, 270190879, 433773933, 693518984

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Partial sums of A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
Index entries for sequences related to partitions

Cf. A000041, A000070, A006906, A091360, A302831.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

nmax = 35; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j).
From Vaclav Kotesovec, Apr 14 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 319343.48587983201292657132469068725642363369445... if mod(n,3)=0
c = 319343.34569378454521307030620964478962032866022... if mod(n,3)=1
c = 319343.21458897980925594955657564398036486423380... if mod(n,3)=2
(End)

A091357 Number of planar partitions of n with exactly 3 rows.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 78, 138, 239, 405, 669, 1088, 1741, 2744, 4267, 6564, 9975, 15019, 22394, 33111, 48549, 70678, 102127, 146636, 209186, 296697, 418401, 586985, 819218, 1137962, 1573336, 2165888, 2968914, 4053563, 5512820, 7469989

Offset: 3

Christian G. Bower, Jan 02 2004

nonn

Column 3 of A091355.

Cf. A000219, A089924, A091356-A091360.

a(n) = A000991(n)-A000990(n).

A276432 Sum of the traces of all plane partitions of n.

Original entry on oeis.org

1, 4, 10, 26, 56, 126, 252, 512, 980, 1866, 3427, 6258, 11121, 19618, 33975, 58328, 98732, 165804, 275246, 453544, 740338, 1200088, 1929897, 3083898, 4893775, 7720826, 12106814, 18883104, 29291740, 45215386, 69451631, 106197524, 161656759, 245050410, 369935066

Offset: 1

Emeric Deutsch, Sep 24 2016

nonn

Convolution of A000203 and A000219. - Vaclav Kotesovec, Sep 25 2016

Convolution of A340793 and A091360. - Omar E. Pol, Feb 16 2021

			a(3) = 10 because the 6 (=A000219(3)) planar partitions of 3 are [3], [2,1], [2;1], [1,1,1], [1;1;1], [1,1;1] (; indicates a new row); the sum of their traces is 3+2+2+1+1+1 = 10.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 179-201.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000219, A089353, A277029.

Cf. A091360, A340793.

g:= (sum(j*x^j/(1-x^j),j = 1..100))/(product((1-x^k)^k,k = 1..100)): gser := series(g, x = 0,40): seq(coeff(gser, x, m), m = 1 .. 35);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p
      ->p+[0, j*p[1]])(b(n-i*j, i-1))*binomial(i+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018

nmax = 50; Rest[CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 25 2016 *)

G.f.: g(x) = Sum_{j>=1} (j*x^j/(1-x^j))/Product_{k>=1} (1-x^k)^k.

a(n) = Sum(k*A089353(n,k), k>=1).

A302832 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^k)^k.

Original entry on oeis.org

1, 2, 4, 9, 17, 33, 61, 110, 193, 335, 570, 955, 1582, 2586, 4185, 6706, 10646, 16757, 26178, 40587, 62503, 95637, 145445, 219929, 330766, 494898, 736858, 1092027, 1611185, 2367079, 3463490, 5048009, 7329935, 10605211, 15290942, 21973641, 31475620, 44946859, 63991639, 90842560

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Partial sums of A026007.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000009, A026007, A036469, A091360, A302831.

b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n))
    end:
g:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+g(n)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

nmax = 39; CoefficientList[Series[1/(1 - x) Product[(1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[1/(1 - x) Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ exp((3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/12) * 3^(2/3) * sqrt(Pi) * Zeta(3)^(1/6) * n^(1/3)).
a(n) ~ (2*n/(3*Zeta(3)))^(1/3) * A026007(n).
a(n) ~ erfi((3/2)^(2/3) * Zeta(3)^(1/6) * n^(1/3)) / 2^(13/12).
(End)

A091358 Number of planar partitions of n with exactly 4 rows.

Original entry on oeis.org

1, 2, 5, 11, 24, 46, 90, 164, 298, 520, 899, 1512, 2521, 4116, 6659, 10609, 16753, 26126, 40419, 61889, 94067, 141736, 212123, 315087, 465162, 682188, 994857, 1442340, 2080332, 2984724, 4262018, 6056849, 8569913, 12072770, 16938556

Offset: 4

Christian G. Bower, Jan 02 2004

nonn

Column 4 of A091355. Cf. A000219, A091356-A091360.

a(n) = A002799(n)-A000991(n).

A091359 Number of planar partitions of n with exactly 5 rows.

Original entry on oeis.org

1, 2, 5, 11, 24, 48, 94, 176, 324, 580, 1020, 1757, 2985, 4990, 8237, 13428, 21651, 34540, 54583, 85473, 132730, 204484, 312695, 474814, 716217, 1073558, 1599568, 2369781, 3491812, 5118490, 7465789, 10837964, 15661666, 22533586, 32284480

Offset: 5

Christian G. Bower, Jan 02 2004

nonn

Column 5 of A091355. Cf. A000219, A091356-A091360.

a(n) = A001452(n)-A002799(n).

cp's OEIS Frontend

A340793 Sequence whose partial sums give A000203.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

Formula

Extensions

A091356 Number of planar partitions of n with exactly 2 rows.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A302830 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - k*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A091357 Number of planar partitions of n with exactly 3 rows.

Views

Author

Keywords

Crossrefs

Formula

A276432 Sum of the traces of all plane partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A302832 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^k)^k.

Views

Author

Keywords

Comments

A302830 Expansion of (1/(1 - x))Product_{k>=1} 1/(1 - kx^k).