A097788 -id:A097788 - cp's OEIS frontend

A368530 a(n) = Sum_{k=1..n} k^3 * 4^(n-k).

0, 1, 12, 75, 364, 1581, 6540, 26503, 106524, 426825, 1708300, 6834531, 27339852, 109361605, 437449164, 1749800031, 6999204220, 27996821793, 111987293004, 447949178875, 1791796723500, 7167186903261, 28668747623692, 114674990506935, 458699962041564

Offset: 0

Seiichi Manyama, Dec 29 2023

Index entries for linear recurrences with constant coefficients, signature (8,-22,28,-17,4).

Cf. A000578, A000537, A213575, A066999.
Cf. A097788, A368525.
Cf. A014825, A368529.

PARI
```
a(n) = sum(k=1, n, k^3*4^(n-k));
```

G.f.: x * (1+4*x+x^2)/((1-4*x) * (1-x)^4).
a(n) = 8*a(n-1) - 22*a(n-2) + 28*a(n-3) - 17*a(n-4) + 4*a(n-5).
a(n) = (11*4^(n+1) - (9*n^3 + 36*n^2 + 60*n + 44))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^3.

A368535 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^(n-k).

Original entry on oeis.org

0, 1, 6, 31, 188, 1510, 16106, 217938, 3577624, 68952495, 1524157870, 37974983321, 1052320304212, 32089921353308, 1067586804710258, 38470738234990580, 1492501011869912496, 62015249735222969325, 2747431806313734355830, 129267455591507496073315

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A062805, A368534.

Cf. A002663, A097786, A097788.

a(n) = sum(k=1, n, binomial(k+2, 3)*n^(n-k));

a(n) = [x^n] x/((1-n*x) * (1-x)^4).

a(n) = n * (6*n^(n+2) - n^5 - 3*n^4 + n^3 + n^2 - 6*n + 2)/(6 * (n-1)^4) for n > 1.

A097789 a(n)=4a(n-1)+C(n+4,4),n>0, a(0)=1.

Original entry on oeis.org

1, 9, 51, 239, 1026, 4230, 17130, 68850, 275895, 1104295, 4418181, 17674089, 70698176, 282795084, 1131183396, 4524737460, 18098954685, 72395824725, 289583306215, 1158333233715, 4633332945486, 18533331794594, 74133327193326

Offset: 0

Paul Barry, Aug 24 2004

Partial sums of A097788.

Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

G.f. : 1/((1-4x)(1-x)^5); a(n)=4^(n + 5)/243-(27n^4+414n^3+2385n^2+6198n+6248)/1944; a(n)=sum{k=0..n, binomial(n+5, k+5)3^k}.

A229702 Expansion of 1/((1-x)^4*(1-6x)).

Original entry on oeis.org

1, 10, 70, 440, 2675, 16106, 96720, 580440, 3482805, 20897050, 125382586, 752295880, 4513775735, 27082654970, 162495930500, 974975583816, 5849853503865, 35099121024330, 210594726147310, 1263568356885400, 7581410141314171

Offset: 0

Yahia Kahloune, Sep 27 2013

This sequence was chosen to illustrate a way to match generating functions and closed-form solutions.
The general term associated with the generating function
1/((1-s*x)^4*(1-r*x)) with r>s>=1 is a(n) = [ r^(n+4) - s^(n+1)*(s^3 + s^2*(r-s)*binomial(n+4,1) + s*(r-s)^2*binomial(n+4,2)+(r-s)^3*binomial(n+4,3))]/(r-s)^4.

			a(3) = (6^8 - (125*3^3  + 1200*3^2 + 3805*3 + 4026))/3750 = 440.

Index entries for linear recurrences with constant coefficients, signature (10,-30,40,-25,6).

Cf. A002663, A097786, A097788, A097790.

a(n) = (6^(n+4) - (1 + 5*C(n+4,1) + 25*C(n+4,2) + 125*C(n+4,3)))/625 = (6^(n+5) - (125*n^3 + 1200*n^2 + 3805*n + 4026))/3750.

cp's OEIS Frontend

A368530 a(n) = Sum_{k=1..n} k^3 * 4^(n-k).

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A368535 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^(n-k).

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A097789 a(n)=4a(n-1)+C(n+4,4),n>0, a(0)=1.

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A229702 Expansion of 1/((1-x)^4*(1-6x)).

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