A066999 -id:A066999 - cp's OEIS frontend

A047520 a(n) = 2*a(n-1) + n^2, a(0) = 0.

0, 1, 6, 21, 58, 141, 318, 685, 1434, 2949, 5998, 12117, 24378, 48925, 98046, 196317, 392890, 786069, 1572462, 3145285, 6290970, 12582381, 25165246, 50331021, 100662618, 201325861, 402652398, 805305525, 1610611834, 3221224509

Offset: 0

Henry Bottomley, Jul 04 2000

Convolution of squares (A000290) and powers of 2 (A000079). - Graeme McRae, Jun 07 2006
Antidiagonal sums of the convolution array A213568. - Clark Kimberling, Jun 18 2012
This is the partial sums of A050488. - J. M. Bergot, Oct 01 2012
From Peter Bala, Nov 29 2012: (Start)
This is the case m = 2 of the recurrence a(n) = m*a(n-1) + n^m, m = 1,2,..., with a(0) = 0.
The recurrence has the solution a(n) = m^n*Sum_{i=1..n} i^m/m^i and has the o.g.f. A(m,x)/((1-m*x)*(1-x)^(m+1)), where A(m,x) denotes the m-th Eulerian polynomial of A008292.
For other cases see A000217 (m = 1), A066999 (m = 3) and A067534 (m = 4).
(End)
Convolution of A000225 with A005408. - J. M. Bergot, Sep 19 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000295. A000217, A008292, A066999, A067534.

List([0..30], n-> 6*2^n -(n^2+4*n+6)); # G. C. Greubel, Jul 25 2019

a047520 n = sum $ zipWith (*)
                  (reverse $ take n $ tail a000290_list) a000079_list
-- Reinhard Zumkeller, Nov 30 2012

[ 6*2^n-n^2-4*n-6: n in [0..30]]; // Vincenzo Librandi, Aug 22 2011

RecurrenceTable[{a[0]==0,a[n]==2a[n-1]+n^2},a[n],{n,30}] (* or *) LinearRecurrence[{5,-9,7,-2},{0,1,6,21},31] (* Harvey P. Dale, Aug 21 2011 *)
f[n_]:= 2^n*Sum[i^2/2^i, {i, n}]; Array[f, 30] (* Robert G. Wilson v, Nov 28 2012 *)

vector(30, n, n--; 6*2^n -(n^2+4*n+6)) \\ G. C. Greubel, Jul 25 2019

[6*2^n -(n^2+4*n+6) for n in (0..30)] # G. C. Greubel, Jul 25 2019

a(n) = 6*2^n - n^2 - 4*n - 6 = 6*A000225(n) - A028347(n+2).
a(n) = 2^n*Sum_{i=1..n} i^2 / 2^i. - Benoit Cloitre, Jan 27 2002
a(0)=0, a(1)=1, a(2)=6, a(3)=21, a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - Harvey P. Dale, Aug 21 2011
G.f.: x*(1+x)/((1-x)^3*(1-2*x)). - Harvey P. Dale, Aug 21 2011
a(n) = Sum_{k=0..n-1} A000079(n-k) * A000290(k). - Reinhard Zumkeller, Nov 30 2012
E.g.f.: 6*exp(2*x) -(6 +5*x +x^2)*exp(x). - G. C. Greubel, Jul 25 2019

A067534 a(n) = 4^n * Sum_{i=1..n} i^4/4^i.

Original entry on oeis.org

1, 20, 161, 900, 4225, 18196, 75185, 304836, 1225905, 4913620, 19669121, 78697220, 314817441, 1259308180, 5037283345, 20149198916, 80596879185, 322387621716, 1289550617185, 5158202628740, 20632810709441

Offset: 1

Benoit Cloitre, Jan 27 2002

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

Cf. A047520, A066999.

Table[4^n*Sum[i^4/4^i,{i,n}], {n,30}] (* or *) LinearRecurrence[ {9,-30,50,-45,21,-4}, {1,20,161,900,4225,18196}, 30] (* Harvey P. Dale, Jul 15 2012 *)

a(n) = 1/81 * (380*4^n - 27*n^4 - 144*n^3 - 360*n^2 - 528*n - 380). - Ralf Stephan, May 08 2004
a(1)=1, a(2)=20, a(3)=161, a(4)=900, a(5)=4225, a(6)=18196, a(n)= 9*a(n-1)- 30*a(n-2)+50*a(n-3)-45*a(n-4)+21*a(n-5)-4*a(n-6). - Harvey P. Dale, Jul 15 2012
From Peter Bala, Nov 29 2012: (Start)
Recurrence equation: a(n) = 4*a(n-1) + n^4. See A047520 and A066999.
O.g.f.: (x + 11*x^2 + 11*x^3 + x^4)/((1 - 4*x)*(1 - x)^5) = x + 20*x^2 + 161*x^3 + .... (End)

A218376 a(n) = 5^n*sum_{i=1..n} i^5/5^i.

Original entry on oeis.org

0, 1, 37, 428, 3164, 18945, 102501, 529312, 2679328, 13455689, 67378445, 337053276, 1685515212, 8427947353, 42140274589, 210702132320, 1053511710176, 5267559970737, 26337801743253, 131689011192364, 658445059161820

Offset: 0

Robert G. Wilson v, Nov 28 2012

Cf. A000217, A047520, A066999, A067534. A008292.

f[n_] := 5^n*Sum[i^5/5^i, {i, n}]; Array[f, 30, 0]

From Peter Bala, Nov 29 2012: (Start)
a(n) = 1/512*(3535*5^n - (128*n^5 + 800*n^4 + 2400*n^3 + 4600*n^2 + 5700*n + 3535)).
Recurrence equation: a(n) = 5*a(n-1) + n^5.
G.f.: (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)/((1 - 5*x)*(1 - x)^6) = x + 37*x^2 + 428*x^3 + ....
(End)

A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 11, 21, 10, 1, 0, 1, 20, 60, 58, 15, 1, 0, 1, 37, 161, 244, 141, 21, 1, 0, 1, 70, 428, 900, 857, 318, 28, 1, 0, 1, 135, 1149, 3164, 4225, 2787, 685, 36, 1, 0, 1, 264, 3132, 10990, 18945, 18196, 8704, 1434, 45, 1

Offset: 0

Seiichi Manyama, Dec 27 2023

			Square array begins:
  1,  0,   0,    0,     0,      0,      0, ...
  1,  1,   1,    1,     1,      1,      1, ...
  1,  3,   6,   11,    20,     37,     70, ...
  1,  6,  21,   60,   161,    428,   1149, ...
  1, 10,  58,  244,   900,   3164,  10990, ...
  1, 15, 141,  857,  4225,  18945,  81565, ...
  1, 21, 318, 2787, 18196, 102501, 536046, ...

OEIS Wiki, Eulerian polynomials.

Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.
Main diagonal gives A368505.
Cf. A368486.

PARI
```
T(n, k) = sum(j=0, n, k^(n-j)*j^k);
```

G.f. of column k: x*A_k(x)/((1-k*x) * (1-x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

T(0,k) = 0^k; T(n,k) = k*T(n-1,k) + n^k.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A047520 a(n) = 2*a(n-1) + n^2, a(0) = 0.

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A067534 a(n) = 4^n * Sum_{i=1..n} i^4/4^i.

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A218376 a(n) = 5^n*sum_{i=1..n} i^5/5^i.

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A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.

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

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