A213820 -id:A213820 - 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.

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

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

Cf. A212500

b[n_]:=n;c[n_]:=3n-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}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

Original entry on oeis.org

2, 6, 8, 8, 17, 14, 19, 27, 25, 15, 33, 16, 30, 43, 45, 18, 55, 24, 43, 55, 43, 22, 92, 43, 35, 68, 69, 25, 107, 34, 80, 56, 48, 61, 130, 32, 45, 65, 119, 29, 113, 36, 72, 154, 64, 27, 178, 69, 87, 74, 85, 24, 197, 97, 145, 91, 51, 37, 182, 39, 54, 189, 203, 82, 173

Offset: 2

M. F. Hasler, Nov 07 2019

nonn

For n = 1, any (x_1) would be a solution, therefore the offset is 2.
For any n, the equation would also admit the trivial zero solution (0,...,0). If any x_k = 0, then all x_i must be zero, so there is no other solution in the nonnegative integers.
This sequence gives row sums of A328911 which lists the number of solutions with given number of components > 1.
The Shiu paper gives different values for f(2), f(3) and f(8) than those given here, but the author has confirmed (personal communication) that 2, 6 and 19 are correct.
The multiset [1^(n-2),n+1,n*(2*n-1)] is the solution with maximum product. - Hugo Pfoertner, Nov 08 2019
From David A. Corneth, Nov 08 2019: (Start)
Given x_1,...,x_{n-1}, one can find the value x_n by solving the resulting linear univariate equation.
For example, for n = 4, if we are given (1, 1, 7, x_4) then we can solve 4*(9 + x_4) = 7*x_4, getting 36 = 3*x_4, i.e., x_4 = 12. As 12 is an integer and >= x_3 = 7, we have a new solution: (1, 1, 7, 12). (End)
In any solution, we have x_1*...*x_{n-1} <= n^2, implying that a(n) is finite for all n > 1. Furthermore, x_1 = x_2 = ... = x_k = 1 for k = n - 1 - floor(2*log_2(n)). - Max Alekseyev, Nov 10 2019

			For n = 2 variables, we have the equation x1*x2 = 2*(x1 + x2) with positive integer solutions (3,6) and (4,4).
For n = 3, we have 3 solutions in C_1(3) = {(1, 4, 15), (1, 5, 9), (1, 6, 7)} (with 2 components > 1), and 3 others in C_2(3) = {(2, 2, 12), (2, 3, 5), (3,3,3)} (with 3 components > 1), for a total of a(3) = 6.
For n = 4 we have the 8 solutions (1, 1, 5, 28), (1, 1, 6, 16), (1, 1, 7, 12), (1, 1, 8, 10), (1, 2, 3, 12), (1, 2, 4, 7), (1, 3, 4, 4) and (2, 2, 2, 6).
For n = 5, the solutions are, omitting initial components x_i = 1: {(6, 45), (7, 25), (9, 15), (10, 13), (2, 3, 35), (2, 5, 9), (3, 3, 10), (3, 5, 5)}.
For n = 6, the solutions are (omitting x_i = 1): {(7, 66), (8, 36), (9, 26), (10, 21), (11, 18), (12, 16), (2, 4, 27), (2, 5, 15), (2, 6, 11), (2, 7, 9), (3, 3, 18), (3, 4, 10), (3, 6, 6), (2, 2, 2, 24), (2, 2, 3, 9), (2, 2, 4, 6), (2, 3, 3, 5)}.
For n = 9, the 27 solutions are (omitting '1's): {(10, 153), (11, 81), (12, 57), (13, 45), (15, 33), (17, 27), (18, 25), (21, 21), (2, 5, 117), (2, 6, 42), (2, 7, 27), (2, 9, 17), (2, 12, 12), (3, 4, 39), (3, 5, 21), (3, 6, 15), (3, 7, 12), (3, 9, 9), (4, 4, 18), (5, 5, 9), (6, 6, 6), (2, 2, 3, 36), (2, 2, 6, 9), (2, 3, 3, 13), (3, 3, 3, 7), (3, 3, 4, 5), (2, 3, 3, 3, 3)}.

Lars Blomberg, Posting to the Sequence Fans Mailing List, Nov 07 2019, seems to have been the first person to notice that there were problems with the published values (given in A328980). - N. J. A. Sloane, Nov 08 2019
R. K. Guy, Sum equals product. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter D24, (2004), 299-301 (citing Erdős's question dated Aug 19, 1996)

David A. Corneth, Table of n, a(n) for n = 2..6250
David A. Corneth, List of solutions and product of variables of solutions for n = 2..160, omitting ones
Peter Shiu, On Erdős's Last Equation, Amer. Math. Monthly, 126 (2019), 802-808; correction, 127:5 (2020), 478.

Row sums of A328911.

Cf. A000384, A002414, A213820 (apparently maximum product occurring in solutions), A328980, A329205, A329206.

a(n,show=1)={my(s=0,d);forvec(x=vector(n-1,i,[1,n\(sqrt(2)-1)]), 0<(d=vecprod(x)-n) && n*vecsum(x)%d==0 && n*vecsum(x)\d >= x[n-1] &&s++ &&show &&printf("%d,",concat(x,n*vecsum(x)\d)),1);s}

{ A328910(n,k=n-1,m=n^2,p=1,s=0,y=1) = if(k==0, return( p>n && Mod(n*s,p-n)==0 && n*s>=(p-n)*y ) ); sum(x=y, sqrtnint(m,k), A328910(n,k-1,m\x,p*x,s+x,x) ); } \\ Max Alekseyev, Nov 10 2019

More terms from David A. Corneth, Nov 07 2019

A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.

Original entry on oeis.org

1, 2, 36, 300, 1960, 11340, 60984, 312312, 1544400, 7438860, 35103640, 162954792, 746347056, 3380195000, 15164074800, 67476121200, 298135873440, 1309153089420, 5717335239000, 24847720451400, 107520292479600, 463440029892840, 1990477619679120, 8521600803066000

Offset: 0

Alois P. Heinz, May 20 2016

a(0) = 1 by convention.

			a(1) = 2: 12, 21.
a(2) = 36: 1122, 1133, 1144, 1212, 1221, 1313, 1331, 1414, 1441, 2112, 2121, 2211, 2233, 2244, 2323, 2332, 2424, 2442, 3113, 3131, 3223, 3232, 3311, 3322, 3344, 3434, 3443, 4114, 4141, 4224, 4242, 4334, 4343, 4411, 4422, 4433.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A002414, A006752, A213820, A245687.

a:= proc(n) option remember; `if`(n<2, 2^n,
       2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)))
    end:
seq(a(n), n=0..30);

a[n_] := (2*n^3 + n^2 - n) * CatalanNumber[n]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Mar 12 2023 *)

G.f.: 1+(8*x+1)*2*x/(1-4*x)^(5/2).
a(n) = C(2*n,n)*C(2*n,2) for n>0, a(0)=1.
a(n) = 2*C(2*(n-1),n-1)*(2*n-1)^2, a(0)=1.
a(n) = 2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)) for n>1, a(n) = 2^n for n=0..1.
a(n) = A245687(2n,n).
a(n) = A000108(n)*A213820(n) = 2*A000108(n)*A002414(n) for n>0, a(0)=1.
Sum_{n>=0} 1/a(n) = 1 - log(sqrt(3)+2)*Pi/6 + 4*G/3, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 12 2023

A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

Original entry on oeis.org

1, 2, 3, 6, 18, 6, 19, 158, 60, 10, 63, 1611, 916, 140, 15, 216, 17811, 16698, 3060, 270, 21, 760, 207395, 336210, 81090, 7690, 462, 28, 2725, 2505858, 7218768, 2396434, 268005, 16226, 728, 36, 9910, 31125711, 162185112, 76020890, 10477161, 701589, 30408, 1080, 45

Offset: 2

John Mason, Jul 07 2025

			The top corner of the array (size on horizontal axis, dimensions on vertical):
              1    2     3       4         5          6           7           8         9         10
(A001168) 2:  1    2     6      19        63        216         760        2725      9910      36446
(A075678) 3:  3   18   158    1611     17811     207395     2505858    31125711 394982973 5098498323
(A366335) 4:  6   60   916   16698    336210    7218768   162185112  3769221330
          5: 10  140  3060   81090   2396434   76020890  2535403620 87781527395
          6: 15  270  7690  268005  10477161  441378400 19603138320
          7: 21  462 16226  701589  34160301 1796996509
          8: 28  728 30408 1570436  91583156
          9: 36 1080 52296 3141108 213477012

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Rows: A001168 (n=2), A075678 (n=3), A366335 (n=4).
Columns: A000217 (k=1), A213820 (k=2).
Cf. A385291 (polyominoes), A385581 (polysticks).

A277977 a(n) = n(1-3n+2n^2+2*n^3)/2.

Original entry on oeis.org

0, 1, 19, 96, 298, 715, 1461, 2674, 4516, 7173, 10855, 15796, 22254, 30511, 40873, 53670, 69256, 88009, 110331, 136648, 167410, 203091, 244189, 291226, 344748, 405325, 473551, 550044, 635446, 730423, 835665, 951886, 1079824, 1220241, 1373923, 1541680, 1724346

Offset: 0

Emeric Deutsch, Nov 07 2016

For n>=3, a(n) is the second Zagreb index of the graph obtained by joining one vertex of a complete graph K[n] with each vertex of a second complete graph K[n].

The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.

			a(4) = 298. Indeed, the corresponding graph has 16 edges. We list the degrees of their endpoints: (3,3), (3,3), (3,3), (3,7), (3,7), (3,7), (4,4), (4,4), (4,4), (4,4), (4,4), (4,4), (4,7), (4,7), (4,7), (4,7). Then, the second Zagreb index is 3*9 + 3*21 + 6*16 + 4*28 = 298.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A213820.

seq((1/2)*n*(1-3*n+2*n^2+2*n^3), n = 0 .. 45);

a(n) = n*(1-3*n+2*n^2+2*n^3)/2 \\ Felix Fröhlich, Nov 07 2016

concat(0, Vec(x*(1+x)*(1+13*x-2*x^2)/(1-x)^5 + O(x^40))) \\ Felix Fröhlich, Nov 07 2016

G.f.: x*(1+x)*(1+13*x-2*x^2)/(1-x)^5. - Robert Israel, Nov 07 2016

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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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

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A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

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A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.

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A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

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A277977 a(n) = n*(1-3n+2*n^2+2*n^3)/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.

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

A277977 a(n) = n(1-3n+2n^2+2*n^3)/2.