A231308 -id:A231308 - cp's OEIS frontend

A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1.

0, 1, 16, 82, 272, 707, 1568, 3108, 5664, 9669, 15664, 24310, 36400, 52871, 74816, 103496, 140352, 187017, 245328, 317338, 405328, 511819, 639584, 791660, 971360, 1182285, 1428336, 1713726, 2042992, 2421007, 2852992, 3344528, 3901568, 4530449, 5237904

Offset: 0

Stanislav Sykora, Nov 07 2013

In physics, a(n)/2^(M-1) is the trace of the spin operator |S_z|^M for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of their 4th powers is 2*82/16 = a(3)/8 (analogously for other values of M).

Partial sums of A062392. - Bruce J. Nicholson, Jun 29 2019

			a(4) = 4^4 + 2^4 = 272; a(5) = 5^4 + 3^4 + 1^4 = 707.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

Cf. A000389, A000579, A000332, A000583.

List([0..40], n-> n*(3*n^4+15*n^3+20*n^2-8)/30); # G. C. Greubel, Jul 01 2019

[1/30*n*(3*n^4+15*n^3+20*n^2-8): n in [0..40]]; // Vincenzo Librandi, Dec 23 2015

Table[SeriesCoefficient[x*(1+10*x+x^2)/(1-x)^6, {x, 0, n}], {n, 0, 40}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 16, 82, 272, 707}, 40] (* Vincenzo Librandi, Dec 23 2015 *)

nmax=40;a = vector(nmax);a[2]=1;for(i=3,#a,a[i]=a[i-2]+(i-1)^4); print(a);

concat(0, Vec(x*(1+10*x+x^2)/(1-x)^6 + O(x^40))) \\ Colin Barker, Dec 22 2015

[n*(3*n^4+15*n^3+20*n^2-8)/30 for n in (0..40)] # G. C. Greubel, Jul 01 2019

a(n) = Sum_{k=0..floor(n/2)} (n - 2*k)^4.
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/30)*n*(3*n^4 + 15*n^3 + 20*n^2 - 8).
G.f.: x*(1 + 10*x + x^2) / (1-x)^6.
(End)
E.g.f.: x*(30 + 210*x + 185*x^2 + 45*x^3 + 3*x^4)*exp(x)/30. - G. C. Greubel, Apr 24 2016
From Bruce J. Nicholson, Jun 29 2019: (Start)
a(n) = 12*A000389(n+3) + A000292(n);
a(n) = (12*A000579(n+4)+A000332(n+3)) - (12*A000579(n+3)+A000332(n+2));
a(n) - a(n-2) = A000583(n). (End)

A231304 Recurrence a(n) = a(n-2) + n^M for M=5, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 32, 244, 1056, 3369, 8832, 20176, 41600, 79225, 141600, 240276, 390432, 611569, 928256, 1370944, 1976832, 2790801, 3866400, 5266900, 7066400, 9351001, 12220032, 15787344, 20182656, 25552969, 32064032, 39901876, 49274400, 60413025, 73574400, 89042176

Offset: 0

Stanislav Sykora, Nov 07 2013

See A231303.

			a(4) = 4^5 + 2^5 = 1056; a(5) = 5^5 + 3^5 + 1^5 = 3369.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+n^5},a,{n,30}] (* or *) LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{0,1,32,244,1056,3369,8832,20176},40] (* Harvey P. Dale, Jul 22 2014 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^5); print(a);

concat(0, Vec(x*(1+26*x+66*x^2+26*x^3+x^4)/((1-x)^7*(1+x)) + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^5.
a(0)=0, a(1)=1, a(2)=32, a(3)=244, a(4)=1056, a(5)=3369, a(6)=8832, a(7)=20176, a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8). - Harvey P. Dale, Jul 22 2014
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/24)*(2*n^6 + 12*n^5 + 20*n^4 - 16*n^2 - 3*(-1)^n + 3).
G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4) / ((1-x)^7*(1+x)).
(End)

A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 64, 730, 4160, 16355, 50816, 134004, 312960, 665445, 1312960, 2437006, 4298944, 7263815, 11828480, 18654440, 28605696, 42792009, 62617920, 89837890, 126617920, 175604011, 239997824, 323639900, 431100800, 567780525, 740016576, 955201014, 1221906880

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^6 + 3^6 + 1^6 = 16355.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231306 (M=7), A231307 (M=8), A231308 (M=9), A231309 (M=10).

map(op,ListTools:-PartialSums([seq([(2*i)^6,(2*i+1)^6],i=0..50)])); # Robert Israel, Dec 22 2015

Table[SeriesCoefficient[x (1 + 56 x + 246 x^2 + 56 x^3 + x^4)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,64,730,4160,16355,50816,134004},30] (* Harvey P. Dale, Aug 23 2025 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^6); print(a);

concat(0, Vec(x*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^8 + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)}(n-2k)^6.
From Colin Barker, Dec 22 2015: (Start)
a(n) = 1/42*n*(3*n^6+21*n^5+42*n^4-56*n^2+32).
G.f.: x*(1+56*x+246*x^2+56*x^3+x^4) / (1-x)^8.
(End)

A231306 Recurrence a(n) = a(n-2) + n^M for M=7, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 128, 2188, 16512, 80313, 296448, 903856, 2393600, 5686825, 12393600, 25173996, 48225408, 87922513, 153638912, 258781888, 422074368, 669120561, 1034294400, 1562992300, 2314294400, 3364080841, 4808652288, 6768906288, 9395123712, 12872421913, 17426933888

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^7 + 3^7 + 1^7 = 80313.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-27,48,-42,0,42,-48,27,-8,1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231307 (M=8), A231308 (M=9), A231309 (M=10).

Table[SeriesCoefficient[x (1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/((1 - x)^9 (1 + x)), {x, 0, n}], {n, 0, 26}] (* Michael De Vlieger, Dec 22 2015 *)
LinearRecurrence[{8, -27, 48, -42, 0, 42, -48, 27, -8, 1}, {0, 1, 128, 2188, 16512, 80313, 296448, 903856, 2393600, 5686825}, 30] (* Vincenzo Librandi, Dec 23 2015 *)
nxt[{n_,a_,b_}]:={n+1,b,a+(n+1)^7}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Jun 16 2022 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^7); print(a);

concat(0, Vec(x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/((1-x)^9*(1+x)) + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^7.
From Colin Barker, Dec 22 2015: (Start)
a(n) = 1/96*(6*n^8+48*n^7+112*n^6-224*n^4+256*n^2+51*((-1)^n-1)).
G.f.: x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6) / ((1-x)^9*(1+x)).
(End)

A231307 Recurrence a(n) = a(n-2) + n^M for M=8, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 256, 6562, 65792, 397187, 1745408, 6161988, 18522624, 49208709, 118522624, 263567590, 548504320, 1079298311, 2024293376, 3642188936, 6319260672, 10617946377, 17339221248, 27601509418, 42939221248, 65424368779, 97815094784, 143735354060

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^8 + 3^8 + 1^8 = 397187.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231308 (M=9), A231309 (M=10).

[1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192): n in [0..30]]; // Vincenzo Librandi, Dec 23 2015

Table[1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192),{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)

nmax=100; a = vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^8); print(a);

concat(0, Vec(x*(1+246*x+4047*x^2+11572*x^3+4047*x^4+246*x^5+x^6)/(1-x)^10 + O(x^50))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n/2)} (n-2k)^8.
a(n) = 1/90*n*(n+1)*(n+2)*(5*n^6+30*n^5+20*n^4-120*n^3-16*n^2+288*n-192). - Vaclav Kotesovec, Feb 14 2014
G.f.: x*(1+246*x+4047*x^2+11572*x^3+4047*x^4+246*x^5+x^6) / (1-x)^10.

A231309 Recurrence a(n) = a(n-2) + n^M for M=10, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1024, 59050, 1049600, 9824675, 61515776, 292299924, 1135257600, 3779084325, 11135257600, 29716508926, 73052621824, 167575000775, 362307276800, 744225391400, 1461818904576, 2760219291849, 5032286131200, 8891285549650, 15272286131200

Offset: 0

Stanislav Sykora, Nov 07 2013

			a(5) = 5^10 + 3^10 + 1^10 = 9824675.

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A001477 (M=1), A000292 (M=2), A105636 (M=3), A231303 (M=4), A231304 (M=5), A231305 (M=6), A231306 (M=7), A231307 (M=8), A231308 (M=9).

CoefficientList[Series[x (1 + 1012 x + 46828 x^2 + 408364 x^3 + 901990 x^4 + 408364 x^5 + 46828 x^6 + 1012 x^7 + x^8)/(1 - x)^12, {x, 0, 20}], x] (* Michael De Vlieger, Nov 20 2019 *)

nmax=100; a=vector(nmax); a[2]=1; for(i=3, #a, a[i]=a[i-2]+(i-1)^10); print(a);

concat(0, Vec(x*(1 +1012*x +46828*x^2 +408364*x^3 +901990*x^4 +408364*x^5 +46828*x^6 +1012*x^7 +x^8) / (1 -x)^12 + O(x^40))) \\ Colin Barker, Dec 22 2015

a(n) = Sum_{k=0..floor(n\2)} (n-2*k)^10.
From Colin Barker, Dec 22 2015: (Start)
a(n) = (1/66)*n*(3*n^10 + 33*n^9 + 110*n^8 - 528*n^6 + 2112*n^4 - 4224*n^2 + 2560).
G.f.: x*(1 + 1012*x + 46828*x^2 + 408364*x^3 + 901990*x^4 + 408364*x^5 + 46828*x^6 + 1012*x^7 + x^8) / (1-x)^12.
(End)

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A231303 Recurrence a(n) = a(n-2) + n^M for M=4, starting with a(0)=0, a(1)=1.

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A231304 Recurrence a(n) = a(n-2) + n^M for M=5, starting with a(0)=0, a(1)=1.

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A231305 Recurrence a(n) = a(n-2) + n^M for M=6, starting with a(0)=0, a(1)=1.

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A231306 Recurrence a(n) = a(n-2) + n^M for M=7, starting with a(0)=0, a(1)=1.

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A231307 Recurrence a(n) = a(n-2) + n^M for M=8, starting with a(0)=0, a(1)=1.

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A231309 Recurrence a(n) = a(n-2) + n^M for M=10, starting with a(0)=0, a(1)=1.

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