A213568 -id:A213568 - cp's OEIS frontend

A213569 Principal diagonal of the convolution array A213568.

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

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

Original entry on oeis.org

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.

A000247 a(n) = 2^n - n - 2.

Original entry on oeis.org

0, 3, 10, 25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, 65518, 131053, 262124, 524267, 1048554, 2097129, 4194280, 8388583, 16777190, 33554405, 67108836, 134217699, 268435426, 536870881, 1073741792, 2147483615

Offset: 2

N. J. A. Sloane

Ways of placing n+1 labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.
2^a(n) is an integer of the form 1/(2 - Sum_{i=1..m} i/2^i). - Benoit Cloitre, Oct 25 2002
Number of permutations avoiding 13-2 that contain the pattern 23-1 exactly twice.
Cost of ternary maximum height Huffman tree with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n = 2N + 1. - Alex Vinokur (alexvn(AT)barak-online.net), Nov 02 2004
a(n) is the number of Dyck n-paths whose third upstep initiates the last long ascent, n >= 1. A long ascent is one consisting of 2 or more upsteps. For example, a(3)=3 counts UUDuUDDD, UDUDuUDD, UUDDuUDD (third upstep in small type). - David Callan, Dec 08 2004
Subsequence of A158581; A000120(a(n)) > 1. - Reinhard Zumkeller, Apr 16 2009
Number of vertices of the tropical Grassmannian simplicial complex G(2,n), related to phylogenetic trees. - Tom Copeland, Oct 03 2011
(Conjecture) Let a(2)=0. For n > 2, let N = 2*n + 1. For each n, define the n X n tridiagonal unit-primitive matrix (see [Jeffery]) A_{N,1}=[0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1] associated with N. Define the n-dimensional column vectors V_N = [v_1,v_2,...,v_n]^T = [A_{N,1}]^n*[1,1,...,1]^T, where [.]^T denotes matrix transpose and [1,...,1] is the n-dimensional unit vector. Let (v_k)N denote the k-th element of V_N, k in {1,...,n}. Then a(n) = (v(n-2))N. - _L. Edson Jeffery, Jan 20 2012
For n>0, (a(n)) is row 3 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
For n>2, a(n-2) is the number of connected induced (non-null) subgraphs of the n-centipede graph. - Giovanni Resta, May 04 2017
a(n) is the number of maximal boundary strata of the moduli space of stable rational curves with n+1 marked points. The closures of the maximal boundary strata are called the irreducible boundary divisors of the moduli space; see Cavalieri Section 2.1. - Harry Richman, Aug 13 2024

			a(3) = 4!/(2!*2!*2!) = 3.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..300
Renzo Cavalieri, Moduli spaces of pointed rational curves, (2016).
Antal E. Fekete, Apropos Two Notes on Notation, The Amer. Math. Monthly, Vol. 101, No. 8 (Oct., 1994), pp. 771-778. See p. 776.
Robert Israel et al, Primes 2^n - n - 2, Mathematics StackExchange.
L. E. Jeffery, Unit-primitive matrices
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
Mathoverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Alex Vinokur, Fibonacci-like polynomials produced by m-ary Huffman codes for absolutely ordered sequences, arXiv:cs/0411002 [cs.DM], 2004.
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000478 (3 boxes), A058844 (4 boxes).

List([2..40], n-> 2^n -n-2); # G. C. Greubel, Jul 26 2019

[2^n -n-2: n in [2..40]]; // G. C. Greubel, Jul 26 2019

A000247:=(-3+2*z)/((2*z-1)*(z-1)**2); # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{4,-5,2}, {0,3,10}, 40] (* Harvey P. Dale, Aug 23 2011 *)
Table[2^n -n-2, {n,2,40}] (* Eric W. Weisstein, Aug 09 2017 *)

A000247(n):=2^n-n-2$
makelist(A000247(n),n,2,30); /* Martin Ettl, Nov 08 2012 */

a(n)=2^n-n-2 \\ Charles R Greathouse IV, Sep 28 2015

[2^n -n-2 for n in (2..40)] # G. C. Greubel, Jul 26 2019

E.g.f.: (exp(x)-1-x)*(exp(x)-1).
G.f.: x^3*(3-2*x)/((1-2*x)*(1-x)^2).
a(n) = 2*a(n-1) + n + 3 = a(n-1) + 2^(n-1) - 1 = A000295(n) - 1 = A000295(n+1) - 2^(n+1).
A107907(a(n)) = A000225(n). - Reinhard Zumkeller, May 28 2005
Starting (3, 10, 25, 56, ...) = binomial transform of [3, 7, 8, 8, 8, ...]. - Gary W. Adamson, Nov 07 2007
a(2)=0, a(3)=3, a(4)=10, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, Aug 23 2011
a(n) = (Sum_{k=2..floor(n/2)} binomial(n+1,k)) + if(n odd, binomial(n+1,(n+1)/2)/2, 0).
a(n) = Sum_{k=0..n-3} Sum_{i=0..n-1} C(i,k). - Wesley Ivan Hurt, Sep 20 2017

Additional comments from Michael Steyer, Dec 02 2000
More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000
I recently changed the beginning of this sequence so the formulas etc. may need to be adjusted. - N. J. A. Sloane, Jan 24 2006
Formulas and comments adjusted for offset by Franklin T. Adams-Watters, Nov 10 2011

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

Original entry on oeis.org

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

A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.

Original entry on oeis.org

0, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704, 3221225439

Offset: 0

Gary W. Adamson, May 30 2004

A sequence generated from a Bell difference row matrix, companion to A095150.
A095150 uses the same recursion rule but the multiplier [1 1 1] instead of [1 0 0].
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

			a(6) = 183 = 3*88 -2*41 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Rajeev Raman, and Joe Sawada, Generating a Gray code for prefix normal words in amortized polylogarithmic time per word, arXiv:2003.03222 [cs.DS], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000110, A011971, A095149, A095150.

List([0..30], n-> 3*2^n -(n+3)); # G. C. Greubel, Jul 26 2019

[3*2^n -(n+3): n in [0..30]]; // G. C. Greubel, Jul 26 2019

Table[3*2^n -(n+3), {n,0,30}] (* G. C. Greubel, Jul 26 2019 *)

vector(30, n, n--; 3*2^n -(n+3)) \\ G. C. Greubel, Jul 26 2019

[3*2^n -(n+3) for n in (0..30)] # G. C. Greubel, Jul 26 2019

Let M = a 3 X 3 matrix having Bell triangle difference terms (A095149 is composed of differences of the Bell triangle A011971): (fill in the 3 X 3 matrix with zeros): [1 0 0 / 1 1 0 / 2 1 2] = M. Then M^n * [1 0 0] = [1 n a(n)]. E.g. a(4) = 41 since M^4 * [1 0 0] = [1 4 41].
a(n) = 3*2^n -(n+3) = 2*a(n-1) + n +1 = A000295(n+2) - A000079(n). For n>0, a(n) = A077802(n). - Henry Bottomley, Oct 25 2004
From Colin Barker, Apr 23 2012: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: x*(2-x)/((1-x)^2*(1-2*x)). (End)
a(n) = A125128(n) + A000225(n). - Miquel Cerda, Aug 07 2016
a(n) = 2*A125128(n) - A000325(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A125128(n) + A000325(n) + n - 1. - Miquel Cerda, Aug 27 2016
E.g.f.: 3*exp(2*x) - (3+x)*exp(x). - G. C. Greubel, Jul 26 2019
Let Prod_{i=0..n-1} (1+x^{2^i}+x^{2*2^i}) = Sum_{j=0..d} b_j x^j, where d=2^{n+1}-2. Then a(n) = Sum_{j=0..d-1} b_j/b_{j+1} (proved). - Richard Stanley, Aug 27 2019

Edited by Robert G. Wilson v, Jun 05 2004

Deleted a comment and file that were unrelated to this sequence. - N. J. A. Sloane, Aug 17 2025

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

A208638 Number of 3 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

4, 13, 32, 71, 150, 309, 628, 1267, 2546, 5105, 10224, 20463, 40942, 81901, 163820, 327659, 655338, 1310697, 2621416, 5242855, 10485734, 20971493, 41943012, 83886051, 167772130, 335544289, 671088608, 1342177247, 2684354526, 5368709085

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 3 of A208637. Possibly row 4 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
From Noah Carey, Aug 31 2021: (Start)
Conjecture: a(n) is equal to half the sum along the edges of (centered, height 2, width n, starting at line n+1) rectangles in Pascal's triangle, as shown here for n=3 (not including the terms inside the rectangles):
                1
              1   1
            1   2   1     a(3) = (4+6+4 + 15+20+15)/2
          1   3   3   1
        1   4---6---4   1
      1   5 |       | 5   1
    1   6  15--20--15   6   1
  1   7  21  35  35  20   7   1 (End)

			Some solutions for n=4:
  0 1 0 1     0 0 1 0     0 1 0 0     0 0 0 1     0 0 0 0
  0 1 0 0     1 0 1 0     0 1 1 1     1 1 0 0     1 1 1 0
  1 0 1 0     1 0 1 0     1 0 0 1     0 1 1 1     0 0 1 1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A208637.

Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: x*(4 - 3*x) / ((1 - x)^2*(1 - 2*x)).
a(n) = 5*2^n - n - 5.
(End)

cp's OEIS Frontend

A213569 Principal diagonal of the convolution array A213568.

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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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A000247 a(n) = 2^n - n - 2.

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

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A095151 a(n+3) = 3*a(n+2) - 2*a(n+1) + 1 with a(0)=0, a(1)=2.

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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.

A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.