A089658 -id:A089658 - cp's OEIS frontend

A089664 a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

0, 4, 41, 306, 1966, 11540, 63726, 336700, 1720364, 8562024, 41718190, 199753004, 942561636, 4392660376, 20253510956, 92519626200, 419201709976, 1885719209936, 8428262686254, 37453751742604, 165575219275700, 728534225415864, 3191850894862564

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Sequences of S2(n, t): A003583 (t=0), this sequence (t=1), A089665 (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

Table[(n*(3*n+5)*4^n -2*n*(n-1)*Binomial[2*n,n])/8, {n,0,40}] (* G. C. Greubel, May 25 2022 *)

a(n)=n*(3*n+5)*2^(2*n-3) - 3*binomial(n+1,3)*binomial(2*n,n)/(n+1)/2 \\ Charles R Greathouse IV, Oct 23 2023

[(1/2)*(n*(3*n+5)*4^(n-1) -3*binomial(n+1, 3)*catalan_number(n)) for n in (0..40)] # G. C. Greubel, May 25 2022

a(n) = (1/8)*(n*(3*n+5)*4^n - 2*n*(n-1)*binomial(2*n, n)). (see Wang and Zhang, p. 338)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (1/2)*( n*(3*n+5)*4^(n-1) - 3*binomial(n+1, 3)*Catalan(n) ).
G.f.: x*(4*(1-x) - 3*x*sqrt(1-4*x))/(1-4*x)^3.
E.g.f.: 2*x*(2 + 3*x)*exp(4*x) - (x^2/2)*(3*BesselI(0, 2*x) + 4*BesselI(1, 2*x) + BesselI(2, 2*x))*exp(2*x)). (End)

A089669 a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

Original entry on oeis.org

0, 8, 155, 2286, 29296, 344140, 3807774, 40327280, 413058080, 4120742808, 40242188170, 386141947972, 3650905945872, 34087726136672, 314844824466704, 2880757518523200, 26141327872575616, 235490128979282224, 2107598857648209954, 18752794473550896332

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Sequences of S3(n, t): A007403 (t=0), this sequence (t=1), A089670 (t=2), A089671 (t=3), A089672 (t=4).

Cf. A089658, A089664.

a[n_]:= a[n]= Sum[k*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, May 26 2022 *)

@CachedFunction
def A089669(n): return sum(k*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089669(n) for n in (0..40)] # G. C. Greubel, May 26 2022

a(n) = Sum_{k=0..n} k * (Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022
From Vaclav Kotesovec, May 27 2022: (Start)
Recurrence: (n-3)*(n-2)*(n-1)*(81*n^5 - 1080*n^4 + 5769*n^3 - 15146*n^2 + 19080*n - 9088)*a(n) = (n-3)*(1863*n^7 - 29457*n^6 + 195435*n^5 - 696271*n^4 + 1410606*n^3 - 1569664*n^2 + 815328*n - 103936)*a(n-1) - 12*(1134*n^8 - 20709*n^7 + 162279*n^6 - 708529*n^5 + 1865571*n^4 - 2976218*n^3 + 2709336*n^2 - 1189824*n + 153600)*a(n-2) + 64*(405*n^8 - 7101*n^7 + 53712*n^6 - 228226*n^5 + 590469*n^4 - 934993*n^3 + 856278*n^2 - 392944*n + 65280)*a(n-3) + 256*(n-4)*(n-2)*(2*n - 7)*(81*n^5 - 675*n^4 + 2259*n^3 - 3509*n^2 + 2180*n - 384)*a(n-4).
a(n) ~ 3 * n^2 * 8^(n-1) * (1 - 1/sqrt(Pi*n) + (5/3 - 1/(2*Pi*sqrt(3)))/n). (End)

A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 19, 104, 440, 1600, 5264, 16128, 46848, 130560, 352000, 923648, 2369536, 5963776, 14766080, 36044800, 86900736, 207224832, 489357312, 1145569280, 2660761600, 6136266752, 14060355584, 32027705344, 72561459200, 163577856000, 367068708864, 820204535808

Offset: 0

N. J. A. Sloane, Jan 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Columns : A000012, A000012, A000984, A006480, A008977, A008978.

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), this sequence (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

I:=[0,2,19,104]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016

LinearRecurrence[{8,-24,32,-16}, {0,2,19,104}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

[2^(n-3)*n*(7*n^2 + 12*n + 5)/3 for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = 2^(n-3)*n*(7*n^2 + 12*n + 5)/3. (see Wang and Zhang p. 333)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 3.
G.f.: x*(2 + 3*x)/(1 - 2*x)^4. (End)
E.g.f.: x*(12 + 33*x + 14*x^2)*exp(2*x)/6. - Ilya Gutkovskiy, Jun 21 2016

A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 35, 276, 1522, 6820, 26664, 94640, 312512, 975744, 2913280, 8386048, 23416320, 63724544, 169637888, 443043840, 1137934336, 2879979520, 7194083328, 17761304576, 43390730240, 104997322752, 251881062400, 599482433536, 1416470986752, 3324615065600

Offset: 0

N. J. A. Sloane, Jan 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), this sequence (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

[2^(n-6)*n*(3*n*(7+10*n+5*n^2) -2): n in [0..40]]; // G. C. Greubel, May 24 2022

LinearRecurrence[{10,-40,80,-80,32}, {0,2,35,276,1522}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

[n*(15*n^3+30*n^2+21*n-2)*2^(n-6) for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*2^(n-6). - R. J. Mathar, Sep 16 2009
G.f.: x*(2 + 15*x + 6*x^2 + 2*x^3)/(1-2*x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5) for n > 4. - Chai Wah Wu, Jun 21 2016
E.g.f.: x*(8 + 54*x + 60*x^2 + 15*x^3)*exp(2*x)/4. - Ilya Gutkovskiy, Jun 21 2016

A089661 a(n) = S1(n,4), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

Original entry on oeis.org

0, 2, 67, 764, 5492, 30304, 140672, 577920, 2167680, 7577088, 25037056, 79016960, 240028672, 705961984, 2019713024, 5641535488, 15431565312, 41438281728, 109462880256, 284942925824, 732004876288, 1858158460928, 4665915736064, 11600782163968, 28582042664960

Offset: 0

N. J. A. Sloane, Jan 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), this sequence (t=4), A089662 (t=5), A089663 (t=6).

[2^(n-5)*n*(93*n^4+225*n^3+185*n^2+15*n-38)/15: n in [0..30]]; // Vincenzo Librandi, Jun 22 2016

LinearRecurrence[{12,-60,160,-240,192,-64}, {0,2,67,764,5492,30304}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

[n*(n+1)*(93*n^3 +132*n^2 +53*n -38)*2^(n-5)/15 for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = (1/15)*n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*2^(n-5). (See Wang and Zhang, p. 334)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 12*a(n-1) - 60*a(n-2) + 160*a(n-3) - 240*a(n-4) + 192*a(n-5) - 64*a(n-6) for n > 5.
G.f.: x*(2 + 43*x + 80*x^2 + 24*x^3)/(1 - 2*x)^6. (End)
a(n) = 2^(n-5)*n*(93*n^4 + 225*n^3 + 185*n^2 + 15*n - 38)/15. - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/30)*x*(60 + 885*x + 1930*x^2 + 1155*x^3 + 186*x^4)*exp(2*x). - G. C. Greubel, May 24 2022

A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

Original entry on oeis.org

0, 2, 131, 2172, 20386, 138580, 763824, 3631712, 15470144, 60527232, 221297920, 765580288, 2529498624, 8039103488, 24713744384, 73818562560, 215011065856, 612515381248, 1710842904576, 4695105732608, 12682107944960, 33768108982272, 88748191645696

Offset: 0

N. J. A. Sloane, Jan 04 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), this sequence (t=5), A089663 (t=6).

[2^(n-7)*n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4): n in [0..30]]; // Vincenzo Librandi, Jun 22 2016

LinearRecurrence[{14,-84,280,-560,672,-448,128}, {0,2,131,2172,20386,138580, 763824}, 30] (* Vincenzo Librandi, Jun 22 2016 *)

[2^(n-7)*n*(21*n^5 +61*n^4 +55*n^3 +15*n^2 -28*n +4) for n in (0..40)] # G. C. Greubel, May 24 2022

There is an explicit formula for the sum - see Wang and Zhang, p. 334.
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n > 6.
G.f.: x*(-16*x^5 + 64*x^4 + 422*x^3 + 506*x^2 + 103*x + 2)/(1 - 2*x)^7. (End)
a(n) = 2^(n-7)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4). - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/4)*x*(8 + 246*x + 940*x^2 + 1015*x^3 + 376*x^4 + 42*x^5)*exp(2*x). - G. C. Greubel, May 24 2022

A089663 a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 259, 6284, 77180, 646960, 4235864, 23313408, 112793088, 493969920, 1998346240, 7577934848, 27232132096, 93517705216, 308908943360, 986642513920, 3059995508736, 9247515082752, 27310549696512, 79012328898560, 224396746424320, 626707269681152

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), this sequence (t=6).

Cf. A089664, A089669.

[2^(n-7)*n*(381*n^6+1302*n^5+1302*n^4+420*n^3-707*n^2-378*n+368)/21: n in [0..40]]; // G. C. Greubel, May 24 2022

LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0, 2, 259, 6284, 77180, 646960, 4235864, 23313408}, 40] (* G. C. Greubel, Jun 22 2016 *)

[n*(n+1)*(381*n^5 +921*n^4 +381*n^3 +39*n^2 -746*n +368)*2^(n-7)/21 for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = (1/21)*n*(n+1)*(381*n^5 + 921*n^4 + 381*n^3 + 39*n^2 - 746*n + 368) * 2^(n-7). (See Wang and Zhang, p. 334.)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8) for n > 7.
G.f.: x*(2 + 227*x + 2364*x^2 + 4748*x^3 + 2096*x^4 - 72*x^5)/(1 - 2*x)^8. (End)
a(n) = 2^(n-7)*n*(381*n^6 + 1302*n^5 + 1302*n^4 + 420*n^3 - 707*n^2 - 378*n + 368)/21. - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/42)*x*(84 + 5271*x + 33278*x^2 + 57855*x^3 + 37086*x^4 + 9303*x^5 +
762*x^6)*exp(2*x). - G. C. Greubel, May 24 2022

A089665 a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 73, 788, 6630, 48120, 316526, 1940568, 11284380, 62968560, 339954670, 1786320184, 9176663028, 46248446608, 229285525420, 1120646918000, 5409322603896, 25824570392544, 122086747617198, 572130452101240, 2660063893120900, 12279619924999504, 56318986959592676

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), this sequence (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 2), n = 0..40);

Table[(1/24)*(n*(n+1)*(7*n+5)*4^n -4*(n-1)*(3*n^2-2*n+1)*Binomial[2*n-2, n-1]), {n,0,40}] (* G. C. Greubel, May 25 2022 *)

[(n/6)*((n+1)*(7*n+5)*4^(n-1) -(n-1)*(3*n^2-2*n+1)*catalan_number(n-1)) for n in (0..40)] # G. C. Greubel, May 25 2022

a(n) = (1/24)*n*( (n+1)*(7*n+5)*4^n - 2*(n-1)*(3*n^2 - 2*n + 1)*binomial(2*n, n)/(2*n-1) ). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (n/6)*( (n+1)*(7*n+5)*4^(n-1) - (n-1)*(3*n^2 - 2*n + 1)*Catalan(n-1) ).
G.f.: x*(4*(1+3*x) - x*(3 + 2*x + 4*x^2)*sqrt(1-4*x))/(1-4*x)^4.
E.g.f.: x*(4 + 22*x + 56*x^2/3)*exp(4*x) + (x^2/6)*exp(2*x)*( -(9 + 62*x + 145*x^2 + 84*x^3)*f(x, 0) + (36 + 99*x - 32*x^2 - 84 x^3)*f(x, 1) + (45 + 270*x + 284*x^2 + 48*x^3)*f(x, 2) + x*(109 + 224*x + 78*x^2)*f(x, 3) + x^2*(53 + 36*x)*f(x, 4) + 6*x^3*f(x, 5) ), where f(x, n) = BesselI(n, 2*x). (End)

A089666 a(n) = S2(n,3), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 137, 2136, 23452, 209840, 1640346, 11648224, 76976048, 481048128, 2874897670, 16564931504, 92584313112, 504313834336, 2687067833492, 14045889333120, 72202366588096, 365713117287680, 1828223537042142, 9032706189007888, 44158716127799240, 213826835772518304

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), this sequence (t=3), A089667 (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 3), n = 0..40);

Table[n*(15*n^3+30*n^2+21*n-2)*4^(n-3) -(n-1)^2*n^2*(n+1)*Binomial[2*n,n]/(8*(2*n -1)), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)

[n*(15*n^3+30*n^2+21*n-2)*4^(n-3) - 9*binomial(n+1, 3)^2 * catalan_number(n-1)/(n+1) for n in (0..40)] # G. C. Greubel, May 25 2022

a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - (n-1)^2*n^2*(n+1)*binomial(2*n,  n)/(8*(2*n-1)). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - 9*binomial(n+1, 3)^2 * Catalan(n- 1)/(n+1).
G.f.: x*(4*(1 + 15*x + 12*x^2 + 8*x^3) - 3*x*(1 + 6*x - 6*x^2 + 4*x^3)*sqrt(1-4*x))/(1-4*x)^5. (End)

Name changed by G. C. Greubel, May 25 2022

A089667 a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

Original entry on oeis.org

0, 4, 265, 5984, 85722, 944904, 8771462, 72095520, 541127988, 3785356752, 25032083230, 158102986624, 961123994220, 5656943319664, 32386277835772, 181019819948864, 990793669704552, 5323620638111136, 28137973407708174, 146552649537716992

Offset: 0

N. J. A. Sloane, Jan 04 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), A089666 (t=3), this sequence (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

Table[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) -(n-1)*(15*n^5-99*n^3 + 116*n^2-34*n+6)*CatalanNumber[n-2]), {n,0,40}] (* G. C. Greubel, May 25 2022 *)
CoefficientList[Series[x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*Sqrt[1-4*x] )/(1-4*x)^6, {x,0,35}], x] (* Georg Fischer, Nov 09 2022 *)

[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3+116*n^2-34*n+6)*catalan_number(n-2) ) for n in (0..40)] # G. C. Greubel, May 25 2022

a(n) = (1/480)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^n - 4*n*(n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)) ). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = (1/30)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*Catalan(n-2) ).
G.f.: x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*sqrt(1-4*x) )/(1-4*x)^6. [Typo corrected by Georg Fischer, Nov 09 2022] (End)

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A089664 a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

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A089669 a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

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A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

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A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

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A089661 a(n) = S1(n,4), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

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A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

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A089663 a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

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A089665 a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

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