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.

Previous Showing 11-19 of 19 results.

A229331 Total sum of 9th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 514, 20199, 283370, 2256695, 12637956, 55247745, 202345886, 644749920, 1846772550, 4836548836, 11795957334, 27022021703, 58819382790, 122237638440, 244429962966, 471615005229, 882955864560, 1606698758560, 2853601781340, 4952029001892, 8423307325854
Offset: 0

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Author

Alois P. Heinz, Sep 20 2013

Keywords

Comments

The bivariate g.f. for the partition statistic "sum of 9th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^9}*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

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Crossrefs

Column k=9 of A213191.

Programs

  • Maple
    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^9])(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^9*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
  • Mathematica
    (* 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^9, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

Formula

a(n) = Sum_{k=1..n} A066633(n,k) * k^9.
G.f.: g(x) = (Sum_{k>=1} k^9*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 27648*sqrt(3)/11 * exp(Pi*sqrt(2*n/3)) * n^4. - Vaclav Kotesovec, May 28 2018

A229354 Total sum of n-th powers of parts in all partitions of 3.

Original entry on oeis.org

6, 9, 17, 39, 101, 279, 797, 2319, 6821, 20199, 60077, 179199, 535541, 1602519, 4799357, 14381679, 43112261, 129271239, 387682637, 1162785759, 3487832981, 10462450359, 31385253917, 94151567439, 282446313701, 847322163879, 2541932937197, 7625731702719
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=3 of A213191.

Programs

  • Maple
    a:= n-> 4+2^n+3^n:
seq(a(n), n=0..30);
  • Mathematica
    LinearRecurrence[{6,-11,6},{6,9,17},30] (* Harvey P. Dale, Jun 21 2022 *)

Formula

a(n) = Sum_{k=1..3} A066633(3,k) * k^n.
a(n) = 4 + 2^n + 3^n.
G.f.: -(29*x^2-27*x+6)/((x-1)*(2*x-1)*(3*x-1)).

A229355 Total sum of n-th powers of parts in all partitions of 4.

Original entry on oeis.org

12, 20, 44, 122, 392, 1370, 5024, 18962, 72872, 283370, 1110704, 4377602, 17320952, 68727770, 273267584, 1088189042, 4338210632, 17309402570, 69107683664, 276041741282, 1103001557912, 4408513155770, 17623579686944, 70462912522322, 281757456578792
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=4 of A213191.

Programs

  • Maple
    a:= n-> 7+3*2^n+3^n+4^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..4} A066633(4,k) * k^n.
a(n) = 7 + 3*2^n + 3^n + 4^n.
G.f.: -2*(109*x^3-132*x^2+50*x-6)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)).

A229356 Total sum of n-th powers of parts in all partitions of 5.

Original entry on oeis.org

20, 35, 87, 287, 1119, 4775, 21447, 99407, 470319, 2256695, 10936407, 53384927, 261997119, 1291033415, 6381582567, 31620148847, 156969213519, 780378126935, 3884192631927, 19350690855167, 96473921031519, 481256133809255, 2401840755956487, 11991486019167887
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=5 of A213191.

Programs

  • Maple
    a:= n-> 12+4*2^n+2*3^n+4^n+5^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..5} A066633(5,k) * k^n.
a(n) = 12 + 4*2^n + 2*3^n + 4^n + 5^n.
G.f.: -(1814*x^4-2543*x^3+1262*x^2-265*x+20) / ((x-1) *(2*x-1) *(3*x-1) *(4*x-1) *(5*x-1)).

A229357 Total sum of n-th powers of parts in all partitions of 6.

Original entry on oeis.org

35, 66, 180, 660, 2904, 14196, 73920, 400620, 2229624, 12637956, 72573360, 420738780, 2456635944, 14422057716, 85023813600, 502907704140, 2982460443864, 17724476245476, 105513644666640, 628987635392700, 3753738850485384, 22422625749793236, 134041199563164480
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=6 of A213191.

Programs

  • Maple
    a:= n-> 19+8*2^n+4*3^n+2*4^n+5^n+6^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..6} A066633(6,k) * k^n.
a(n) = 19 + 8*2^n + 4*3^n + 2*4^n + 5^n + 6^n.
G.f.: -(18144*x^5 -28874*x^4 +17295*x^3 -4919*x^2 +669*x -35) / Product_{j=1..6} (j*x-1).

A229358 Total sum of n-th powers of parts in all partitions of 7.

Original entry on oeis.org

54, 105, 311, 1281, 6407, 35745, 212951, 1323441, 8464487, 55247745, 365983991, 2451448401, 16559916167, 112602093345, 769628452631, 5282089330161, 36372360161447, 251135368228545, 1737811434946871, 12047233511096721, 83642479068080327, 581449745964789345
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=7 of A213191.

Programs

  • Maple
    a:= n-> 30+11*2^n+6*3^n+3*4^n+2*5^n+6^n+7^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..7} A066633(7,k) * k^n.
a(n) = 30 + 11*2^n + 6*3^n + 3*4^n + 2*5^n + 6^n + 7^n.
G.f.: -(196356*x^6 -339112*x^5 +230407*x^4 -79457*x^3 +14759*x^2 -1407*x +54) / Product_{j=1..7} (j*x-1).

A229359 Total sum of n-th powers of parts in all partitions of 8.

Original entry on oeis.org

86, 176, 558, 2486, 13578, 83486, 552378, 3835406, 27530298, 202345886, 1513288698, 11466138926, 87752274618, 676845479486, 5252962429818, 40970428516046, 320834049236538, 2520676708888286, 19857791921151738, 156791682937990766, 1240318818550256058
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=8 of A213191.

Programs

  • Maple
    a:= n-> 45+19*2^n+9*3^n+6*4^n+3*5^n+2*6^n+7^n+8^n:
seq(a(n), n=0..30);
  • Mathematica
    LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{86,176,558,2486,13578,83486,552378,3835406},30] (* Harvey P. Dale, Aug 10 2021 *)

Formula

a(n) = Sum_{k=1..8} A066633(8,k) * k^n.
a(n) = 45 + 19*2^n + 9*3^n + 6*4^n + 3*5^n + 2*6^n + 7^n + 8^n.
G.f.: -2*(1213656*x^7 -2263598*x^6 +1707227*x^5 -680514*x^4 +155801*x^3 -20589*x^2 +1460*x -43) / Product_{j=1..8} (j*x-1).

A229360 Total sum of n-th powers of parts in all partitions of 9.

Original entry on oeis.org

128, 270, 910, 4392, 26218, 177120, 1292410, 9924912, 78974938, 644749920, 5365004410, 45294444432, 386724933658, 3331246259520, 28899114836410, 252142576067952, 2210222303020378, 19448976596389920, 171690169932572410, 1519690359242075472, 13481625657159443098
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=9 of A213191.

Programs

  • Maple
    a:= n-> 67+26*2^n+15*3^n+8*4^n+5*5^n+3*6^n+2*7^n+8^n+9^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..9} A066633(9,k) * k^n.
a(n) = 67 + 26*2^n + 15*3^n + 8*4^n + 5*5^n + 3*6^n + 2*7^n + 8^n + 9^n.
G.f.: -2*(16152120*x^8 -31900986*x^7 +26059495*x^6 -11585520*x^5 +3083861*x^4 -505629*x^3 +50060*x^2 -2745*x +64) / Product_{j=1..9} (j*x-1).

A229361 Total sum of n-th powers of parts in all partitions of 10.

Original entry on oeis.org

192, 420, 1494, 7686, 49218, 358710, 2838234, 23736846, 206418978, 1846772550, 16877063274, 156755849406, 1474458820338, 14008019814390, 134151794416314, 1293111887664366, 12531051927887298, 121969635737464230, 1191547891970937354, 11676481763813277726
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2013

Keywords

Links

Crossrefs

Row n=10 of A213191.

Programs

  • Maple
    a:= n-> 97+41*2^n+21*3^n+13*4^n+8*5^n+5*6^n+3*7^n+2*8^n+9^n+10^n:
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=1..10} A066633(10,k) * k^n.
a(n) = 97+41*2^n+21*3^n+13*4^n+8*5^n+5*6^n+3*7^n+2*8^n+9^n+10^n.
G.f.: -6*(79272960*x^9 -165073604*x^8 +144452636*x^7 -70246061*x^6 +21041460*x^5 -4044664*x^4 +500814*x^3 -38639*x^2 +1690*x -32) / Product_{j=1..10} (j*x-1).
Previous Showing 11-19 of 19 results.

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