A047520 -id:A047520 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

Original entry on oeis.org

0, -1, 2, -5, 6, -13, 10, -29, 6, -69, -38, -197, -250, -669, -1142, -2509, -4762, -9813, -19302, -38965, -77530, -155501, -310518, -621565, -1242554, -2485733, -4970790, -9942309, -19883834, -39768509, -79536118

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to keep the values integer.

See also A232600 and references therein for integer values of q.

			a(3) = 2^3 * [0^2/2^0 - 1^2/2^1 + 2^2/2^2 - 3^2/2^3] = -5.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-1,3,5,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232604 (p=3,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232603:= n-> ((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27; seq(A232603(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-1,3,5,2}, {0,-1,2,-5}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*(9*n^2+12*n+2)-2^(n+1))/27;

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(9*n^2+12*n+2) - 2^(n+1))/27.
G.f.: x*(-1+x)/( (1-2*x)*(1+x)^3 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(-2*exp(2*x) + (2 -21*x +9*x^2)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - a(n-1) + 3*a(n-2) + 5*a(n-3) + 2*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

Original entry on oeis.org

0, -1, 6, -15, 34, -57, 102, -139, 234, -261, 478, -375, 978, -241, 2262, 1149, 6394, 7875, 21582, 36305, 80610, 151959, 314566, 616965, 1247754, 2479883, 4977342, 9935001, 19891954, 39759519, 79546038, 159062285

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to make the values integers.
See also A232600 and references therein for integer values of q.
The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - Richard R. Forberg, Feb 17 2014.

			a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-2,2,8,7,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232604:= n-> (2^(n+1) +(-1)^n*(9*n^3 +18*n^2 +6*n -2))/27; seq(A232604(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-2,2,8,7,2}, {0,-1,6,-15,34}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = (2^(n+1) + (-1)^n*(9*n^3+18*n^2+6*n-2))/27.
G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(2*exp(2*x) - (2 +33*x -45*x^2 +9*x^3)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - 2*a(n-1) + 2*a(n-2) + 8*a(n-3) + 7*a(n-4) + 2*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A213568 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213569
Antidiagonal sums:  A047520
Row 1,  (1,3,6,...)**(1,4,9,...):  A125128
Row 2,  (1,3,6,...)**(4,9,16,...):  A095151
Row 3,  (1,3,6,...)**(9,16,25,...):  A000247
Row 4,  (1,3,6,...)**(16,25,36...):  A208638 (?)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
  1...4....11...26....57....120
  2...7....18...41....88....183
  3...10...25...56....119...246
  4...13...32...71....150...309
  5...16...39...86....181...372
  6...19...46...101...212...435

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # G. C. Greubel, Jul 26 2019

[2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Second program *)
Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 26 2019 *)

for(n=1,12, for(k=1,n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ G. C. Greubel, Jul 26 2019

[[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019

T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by Clark Kimberling, Sep 03 2023
G.f. for row n:  f(x)/g(x), where f(x) = n - (n - 1)*x and g(x) = (1 - 2*x)*(1 - x)^2.
T(n,k) = 2^k*(n + 1) - (n + k + 1). - G. C. Greubel, Jul 26 2019

A066999 a(n) = 3^n * Sum_{i=1..n} i^3/3^i.

Original entry on oeis.org

1, 11, 60, 244, 857, 2787, 8704, 26624, 80601, 242803, 729740, 2190948, 6575041, 19727867, 59186976, 177565024, 532699985, 1598105787, 4794324220, 14382980660, 43148951241, 129446864371, 388340605280, 1165021829664

Offset: 1

Benoit Cloitre, Jan 27 2002

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A008292, A047520, A067534.

f[n_] := 3^n*Sum[i^3/3^i, {i, n}]; Array[f, 24] (* Robert G. Wilson v, Nov 28 2012 *)

{ s=0; for (n=1, 200, s+=n^3/3^n; write("b066999.txt", n, " ", 3^n*s) ) } \\ Harry J. Smith, Apr 25 2010

Conjecture: g.f.:(-1-x^2-4*x)/((3*x-1)*(x-1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (11*3^(n+1) - 4*n^3 - 18*n^2 - 36*n - 33)/8. - Vaclav Kotesovec, Nov 28 2012
From Peter Bala, Nov 29 2012: (Start)
Recurrence equation: a(n) = 3*a(n-1) + n^3.
O.g.f.: x*(1 + 4*x + x^2)/((1 - 3*x)*(1 - x)^4) = x + 11*x^2 + 244*x^3 + .... See A047520 and A067534. (End)

A213569 Principal diagonal of the convolution array A213568.

Original entry on oeis.org

1, 7, 25, 71, 181, 435, 1009, 2287, 5101, 11243, 24553, 53223, 114661, 245731, 524257, 1114079, 2359261, 4980699, 10485721, 22020055, 46137301, 96468947, 201326545, 419430351, 872415181, 1811939275, 3758096329, 7784628167

Offset: 1

Clark Kimberling, Jun 18 2012

Create a triangle having first column T(n,1) = 2*n-1 for n = 1,2,3... The remaining terms are set to T(r,c) = T(r,c-1) + T(r-1,c-1). The sum of the terms in row n is a(n). The first five rows of the triangle are 1; 3,4; 5,8,12; 7,12,20,32; 9,16,28,48,80. - J. M. Bergot, Jan 17 2013

Starting at n=1, a(n) = (n+1)*2^n - 2*n - 1. A001787(n) = n*2^n. - J. M. Bergot, Jan 27 2013

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A001787, A213568, A213500.

List([1..30], n-> 2^n*(n+1) -(2*n+1)); # G. C. Greubel, Jul 25 2019

[2^n*(n+1) -(2*n+1): n in [1..30]]; // G. C. Greubel, Jul 25 2019

f:= gfun:-rectoproc({a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4),
  a(1)=1,a(2)=7,a(3)=25,a(4)=71},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 19 2017

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Additional programs *)
LinearRecurrence[{6,-13,12,-4},{1,7,25,71},30] (* Harvey P. Dale, Jan 06 2015 *)
Table[2^n*(n+1) -(2*n+1), {n,30}] (* G. C. Greubel, Jul 25 2019 *)

my(x='x+O('x^30)); Vec(x*(1+x-4*x^2)/((1-2*x)^2*(1-x)^2)) \\ Altug Alkan, Sep 19 2017

vector(30, n, 2^n*(n+1) -(2*n+1)) \\ G. C. Greubel, Jul 25 2019

[2^n*(n+1) -(2*n+1) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: x*(1 + x - 4*x^2)/( (1-2*x)^2*(1-x)^2 ).
a(n) = A001787(n+1)- 2*n - 1. - J. M. Bergot, Jan 22 2013
a(n) = Sum_{k=1..n} Sum_{i=0..n} (n-i) * C(k,i). - Wesley Ivan Hurt, Sep 19 2017

A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2n - 3 + 2h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 21, 14, 5, 58, 43, 22, 7, 141, 110, 65, 30, 9, 318, 255, 162, 87, 38, 11, 685, 558, 369, 214, 109, 46, 13, 1434, 1179, 798, 483, 266, 131, 54, 15, 2949, 2438, 1673, 1038, 597, 318, 153, 62, 17, 5998, 4975, 3442, 2167, 1278, 711, 370, 175, 70

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal: A213757.
Antidiagonal sums: A213758.
Row 1,  (1,3,7,15,31,...)**(1,3,5,7,9,...): A047520.
Row 2,  (1,3,7,15,31,...)**(3,5,7,9,11,...).
Row 3,  (1,3,7,15,31,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....21....58....141...318
3....14...43....110...255...558
5....22...65....162...369...798
7....30...87....214...483...1038
9....38...109...266...597...1278
11...46...131...318...711...1518

Clark Kimberling, Antidiagonals n = 1..40, flattened

Cf. A213500.

b[n_] := -1 + 2^n; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213756 *)
Table[t[n, n], {n, 1, 40}] (* A213757 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213758 *)

T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - 2*x)(1 - x )^3.

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)

A213573 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 6, 4, 21, 17, 9, 58, 50, 34, 16, 141, 125, 93, 57, 25, 318, 286, 222, 150, 86, 36, 685, 621, 493, 349, 221, 121, 49, 1434, 1306, 1050, 762, 506, 306, 162, 64, 2949, 2693, 2181, 1605, 1093, 693, 405, 209, 81, 5998, 5486, 4462, 3310, 2286, 1486

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213574.
Antidiagonal sums: A213575.
row 1, (1,2,4,8,...)**(1,4,9,16...): A047520.
row 2, (1,2,4,8,...)**(4,9,16,25...).
row 3, (1,2,4,8,...)**(9,16,25,36...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
   1,    6,   21,   58,  141,  318, ...
   4,   17,   50,  125,  286,  621, ...
   9,   34,   93,  222,  493, 1050, ...
  16,   57,  150,  349,  762, 1605, ...
  25,   86,  221,  506, 1093, 2286, ...
  36,  121,  306,  693, 1486, 3093, ...
  ...

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*((k+1)^2 +2)- ((n+2)^2 +2) ))); # G. C. Greubel, Jul 25 2019

[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n^2;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[T[n, k], {k, 60}] (* A213573 *)
d = Table[T[n, n], {n, 40}] (* A213574 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
(* Additional programs *)
Table[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2), {n,12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)

for(n=1,12, for(k=1,n, print1(2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2), ", "))) \\ G. C. Greubel, Jul 25 2019

[[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 2019

T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)*x^2 and g(x) = (1 - 2*x)*(1 - x)^3.
T(n,k) = 2^k*(n^2 + 2*n + 3) - (n + k + 2)^2 + 2*(n + k + 1) - 1. - G. C. Greubel, Jul 25 2019

A213574 Principal diagonal of the convolution array A213573.

Original entry on oeis.org

1, 17, 93, 349, 1093, 3093, 8221, 20957, 51861, 125509, 298477, 699789, 1621285, 3718325, 8453181, 19069885, 42728245, 95156901, 210762253, 464517485, 1019214021, 2227173397, 4848613213, 10519312029, 22749902293, 49056576773, 105495131181, 226291086157

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A213568, A213500.

List([1..30], n-> 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)); # G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2): n in [1..30]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Additional programs *)
LinearRecurrence[{9,-33,63,-66,36,-8},{1,17,93,349,1093,3093},30] (* Harvey P. Dale, Jun 25 2014 *)
Rest[CoefficientList[Series[x(1+8x-27x^2+10x^3+16x^4)/(1-3x+2x^2)^3, {x, 0, 30}], x]] (* Vincenzo Librandi, Jun 26 2014 *)

Vec(x*(1+8*x-27*x^2+10*x^3+16*x^4)/((1-x)^3*(1-2*x)^3) + O(x^30)) \\ Colin Barker, Oct 30 2017

vector(30, n, 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)) \\ G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6).
G.f.: x*(1 + 8*x - 27*x^2 + 10*x^3 + 16*x^4)/(1 - 3*x + 2*x^2)^3.
a(n) = 2^n*(3+2*n+n^2) - (3+4*n+4*n^2). - Colin Barker, Oct 30 2017
E.g.f.: (3+6*x+4*x^2)*exp(2*x) - (3+8*x+4*x^2)*exp(x). - G. C. Greubel, Jul 25 2019

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

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A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

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A213568 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A066999 a(n) = 3^n * Sum_{i=1..n} i^3/3^i.

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A213569 Principal diagonal of the convolution array A213568.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A213568 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and = convolution.

A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2n - 3 + 2h, n>=1, h>=1, and ** = convolution.

A213573 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.