A213191 -id:A213191 - cp's OEIS frontend

A066183 Total sum of squares of parts in all partitions of n.

1, 6, 17, 44, 87, 180, 311, 558, 910, 1494, 2302, 3608, 5343, 7986, 11554, 16714, 23549, 33270, 45942, 63506, 86338, 117156, 156899, 209926, 277520, 366260, 479012, 624956, 808935, 1044994, 1340364, 1715572, 2182935, 2770942, 3499379

Offset: 1

Wouter Meeussen, Dec 15 2001

nonn

Sum of hook lengths of all boxes in the Ferrers diagrams of all partitions of n (see the Guo-Niu Han paper, p. 25, Corollary 6.5). Example: a(3) = 17 because for the partitions (3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1}, {3,2,1}, respectively; the total sum of all hook lengths is 6+5+6 = 17. - Emeric Deutsch, May 15 2008
Partial sums of A206440. - Omar E. Pol, Feb 08 2012
Column k=2 of A213191. - Alois P. Heinz, Sep 20 2013
Row sums of triangles A180681, A206561 and A299768. - Omar E. Pol, Mar 20 2018

			a(3) = 17 because the squares of all partitions of 3 are {9}, {4,1} and {1,1,1}, summing to 17.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO], May 09 2008.

Cf. A000041, A001157, A180681, A206440, A206561, A213191, A263004, A265245, A299768.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else g:= b(n, i-1); h:= b(n-i, i);
         [g[1]+h[1], g[2]+h[2] +h[1]*i^2]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2012
# second Maple program:
g := (sum(k^2*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

Table[Apply[Plus, IntegerPartitions[n]^2, {0, 2}], {n, 30}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, g = b[n, i-1]; h = b[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + h[[1]]*i^2}]]; a[n_] :=  b[n, n][[2]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 31 2015, after Alois P. Heinz *)

a(n)=my(s); forpart(v=n,s+=sum(i=1,#v,v[i]^2));s \\ Charles R Greathouse IV, Aug 31 2015

a(n)=sum(k=1,n,sigma(k,2)*numbpart(n-k)) \\ Charles R Greathouse IV, Aug 31 2015

a(n) = Sum_{k=1..n} sigma_2(k)*numbpart(n-k), where sigma_2(k)=sum of squares of divisors of k=A001157(k). - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k>=0} k*A265245(n,k). - Emeric Deutsch, Dec 06 2015
G.f.: g(x) = (Sum_{k>=1} k^2*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 3*sqrt(2)*Zeta(3)/Pi^3 * exp(Pi*sqrt(2*n/3)) * sqrt(n). - Vaclav Kotesovec, May 28 2018

More terms from Naohiro Nomoto, Feb 07 2002

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Original entry on oeis.org

0, 1, 3, 7, 16, 28, 59, 91, 170, 269, 450, 655, 1162, 1602, 2527, 3793, 5805, 8034, 12660, 17131, 26484, 37384, 53738, 73504, 114683, 153613, 221225, 313339, 453769, 609179, 927968, 1223909, 1804710, 2522264, 3539835, 4855420, 7439870, 9765555, 14009545

Offset: 0

Alois P. Heinz, Feb 27 2013

nonn

			a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

From Vaclav Kotesovec, May 24 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0
c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1
c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2
(End)

A229325 Total sum of cubes of parts in all partitions of n.

Original entry on oeis.org

0, 1, 10, 39, 122, 287, 660, 1281, 2486, 4392, 7686, 12628, 20790, 32471, 50694, 76560, 115038, 168333, 245784, 350896, 499620, 699468, 975150, 1341077, 1838550, 2490092, 3361260, 4494084, 5986750, 7909231, 10416300, 13616768, 17745948, 22983345, 29672974

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of cubes of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^3}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Alois P. Heinz, Table of n, a(n) for n = 0..8500
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=3 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^3])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

Table[Total[Flatten[IntegerPartitions[n]^3]],{n,0,40}] (* Harvey P. Dale, May 01 2016 *)
b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], Function[{g, h}, g + h + {0, h[[1]]*i^3}][b[n, i-1], b[n-i, i]]]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^3.
G.f.: g(x) = (Sum_{k>=1} k^3*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n. - Vaclav Kotesovec, May 28 2018

A229332 Total sum of 10th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 1026, 60077, 1110704, 10936407, 72573360, 365983991, 1513288698, 5365004410, 16877063274, 48105808222, 126584890148, 310963328163, 721354362186, 1590587613754, 3359058693214, 6822189191429, 13396265918970, 25501949210562, 47248199227946, 85355336473378

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 10th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^10}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

In general, column k>0 of A213191 is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - Vaclav Kotesovec, May 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=10 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^10])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^10*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^10, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

a(n) = Sum_{j=k..n} A066633(n,k) * k^10.
G.f.: g(x) = (Sum_{k>=1} k^10*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 7054387200*sqrt(2)*Zeta(11)/Pi^11 * exp(Pi*sqrt(2*n/3)) * n^(9/2). - Vaclav Kotesovec, May 28 2018

A252761 Total sum of n-th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 6, 39, 392, 4775, 73920, 1323441, 27530298, 644749920, 16877063274, 486936848068, 15373069624220, 526779275391863, 19477946814752586, 772859962684631760, 32758854443379036238, 1477205045259973740909, 70613293111837146235770, 3566735926987461858837256

Offset: 0

Alois P. Heinz, Dec 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A213191.

Cf. A066633.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
          (b(n-p*m, p-1, k)), m=0..n/p)))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=0..20);

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == 0, l, l + {0, l[[1]]*p^k*m}]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; a[n_] := b[n, n, n][[2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 07 2017, translated from Maple *)

a(n) = Sum_{j=1..n} A066633(n,j) * j^n.

a(n) ~ c * n^n, where c = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131201568288277843252568635840026086375046496... - Vaclav Kotesovec, May 28 2018, updated Jul 21 2018

A229326 Total sum of 4th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 18, 101, 392, 1119, 2904, 6407, 13578, 26218, 49218, 86782, 150860, 249723, 408810, 647170, 1013278, 1545029, 2337738, 3460218, 5086658, 7350874, 10549872, 14929931, 21009874, 29205500, 40385036, 55289000, 75309056, 101692923, 136710130, 182377824

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 4th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^4}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Convolution of A001159 and A000041. - Vaclav Kotesovec, May 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=4 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^4])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^4*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^4, {k, 1, n}]; Array[a, 32, 0] (* Jean-François Alcover, Dec 15 2016 *)
Table[Sum[DivisorSigma[4, k]*PartitionsP[n-k], {k, 1, n}], {n, 0, 40}] (* Vaclav Kotesovec, May 27 2018 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^4.
G.f.: g(x) = (Sum_{k>=1} k^4*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 216*sqrt(2)*Zeta(5)/Pi^5 * exp(Pi*sqrt(2*n/3)) * n^(3/2). - Vaclav Kotesovec, May 28 2018

A229327 Total sum of 5th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 34, 279, 1370, 4775, 14196, 35745, 83486, 177120, 358710, 681316, 1257414, 2212343, 3811590, 6344760, 10381686, 16534989, 25994160, 39973360, 60802860, 90875412, 134507694, 196208405, 283895550, 405646460, 575437476, 807778980, 1126478494, 1556675935

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 5th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^5}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=5 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^5])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^5*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

Table[Total[Flatten[IntegerPartitions[n]]^5],{n,0,30}] (* Harvey P. Dale, Jun 24 2014 *)
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; a[n_] := Sum[T[n, k]*k^5, {k, 1, n}]; Array[a, 45, 0] (* Jean-François Alcover, Dec 15 2016 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^5.
G.f.: g(x) = (Sum_{k>=1} k^5*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 16*sqrt(3)/7 * exp(Pi*sqrt(2*n/3)) * n^2. - Vaclav Kotesovec, May 28 2018

A229328 Total sum of 6th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 66, 797, 5024, 21447, 73920, 212951, 552378, 1292410, 2838234, 5823262, 11464628, 21488403, 39094986, 68600554, 117628414, 196085189, 321067770, 513857202, 810429626, 1254814258, 1918760856, 2889290459, 4305268546, 6331543700, 9226796660, 13297146272

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 6th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^6}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=6 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^6])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^6*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^6, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^6.
G.f.: g(x) = (Sum_{k>=1} k^6*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 38880*sqrt(2)*Zeta(7)/Pi^7 * exp(Pi*sqrt(2*n/3)) * n^(5/2). - Vaclav Kotesovec, May 28 2018

A229329 Total sum of 7th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 130, 2319, 18962, 99407, 400620, 1323441, 3835406, 9924912, 23736846, 52729348, 111173790, 222415631, 428578374, 794363760, 1430855958, 2500747293, 4274146464, 7130355736, 11681752260, 18764913468, 29690460150, 46211761397, 71016916110, 107622522692

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 7th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^7}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=7 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^7])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^7*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^7, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^7.
G.f.: g(x) = (Sum_{k>=1} k^7*x^k/(1-x^k))/Product_{q>=1}(1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 288*sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n^3. - Vaclav Kotesovec, May 28 2018

A229330 Total sum of 8th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 258, 6821, 72872, 470319, 2229624, 8464487, 27530298, 78974938, 206418978, 497450302, 1126668140, 2410089243, 4930872330, 9670306930, 18336637198, 33650085029, 60146112858, 104730757818, 178524931538, 297898015114, 488407670832, 786658901771

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 8th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^8}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=8 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^8])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^8*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^8, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^8.
G.f.: g(x) = (Sum_{k>=1} k^8*x^k/(1-x^k))/Product_{q>=1}(1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 13063680*sqrt(2)*Zeta(9)/Pi^9 * exp(Pi*sqrt(2*n/3)) * n^(7/2). - Vaclav Kotesovec, May 28 2018

cp's OEIS Frontend

A066183 Total sum of squares of parts in all partitions of n.

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A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

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A229325 Total sum of cubes of parts in all partitions of n.

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A229332 Total sum of 10th powers of parts in all partitions of n.

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A252761 Total sum of n-th powers of parts in all partitions of n.

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A229326 Total sum of 4th powers of parts in all partitions of n.

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A229327 Total sum of 5th powers of parts in all partitions of n.

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