A049778 -id:A049778 - 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.

A213849 Rectangular array: (row n) = bc, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 8, 6, 4, 2, 14, 11, 9, 5, 3, 20, 17, 14, 10, 6, 3, 30, 26, 23, 17, 13, 7, 4, 40, 36, 32, 26, 20, 14, 8, 4, 55, 50, 46, 38, 32, 23, 17, 9, 5, 70, 65, 60, 52, 44, 35, 26, 18, 10, 5, 91, 85, 80, 70, 62, 50, 41, 29, 21, 11, 6

Offset: 1

Clark Kimberling, Jul 05 2012

Principal diagonal: A049778.
Antidiagonal sums: A213850.
Row 1, (1,1,2,2,3,3,...)**(1,1,2,2,3,3,...).
Row 2, (1,1,2,2,3,3,...)**(1,2,2,3,3,4,...).
Row 3, (1,1,2,2,3,3,...)**(2,2,3,3,4,4,...).
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80

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

Cf. A212500.

b[n_]:=Floor[(n+1)/2];c[n_]:=Floor[(n+1)/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,1,60}]  (* A213849 *)
d=Table[t[n,n],{n,1,50}] (* A049778 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213850 *)

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*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.

A095800 Triangle T(n,k) = abs( k ( (2n+1)(-1)^(n+k)+2k-1) /4 ) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 1, 4, 2, 2, 9, 2, 6, 3, 16, 3, 4, 12, 4, 25, 3, 8, 6, 20, 5, 36, 4, 6, 15, 8, 30, 6, 49, 4, 10, 9, 24, 10, 42, 7, 64, 5, 8, 18, 12, 35, 12, 56, 8, 81, 5, 12, 12, 28, 15, 48, 14, 72, 9, 100, 6, 10, 21, 16, 40, 18, 63, 16, 90, 10, 121, 6, 14, 15, 32, 20, 54, 21, 80, 18, 110, 11, 144, 7, 12, 24, 20, 45, 24, 70, 24, 99, 20, 132

Offset: 1

Gary W. Adamson, Jun 07 2004

Triangles of increasing sizes are subdivided using a triangular array. Then as shown on p. 83 of Conway and Guy, the series A002717 (1, 5, 13, 27, 48, 78, 118...) denotes the total number of triangles in each figure.
As a conjecture, each row of A095800 could be a distribution governing distinct subsets of types of triangles having the sum in the "How Many Triangles" series A002717. Thus 1 = 1; 5 = (1 + 4), 13 = (2 + 2 + 9)...etc.
Powers of the matrices have alternating signs such that odd rows begin with (+) and even rows begin with (-), as: 1; -1, 4; 2, -2, 9; -2, 6, -3, 16; 3, -4, 12, -4, 25;... Signed row sums = A049778: 1, 3, 9, 17, 32, 48...

			1. [1 0 0 / 1 -2 0 / 1 -2 3]^2 = [1 0 0 / 1 -4 0 / 2 -2 9]. Then change the (-) signs to (+) getting the first 3 rows of the triangle:
1;
1, 4;
2, 2, 9;
2, 6, 3, 16;

J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag New York, 1996, p. 83.

Indranil Ghosh, Rows 1..125, flattened

Cf. A002717 (row sums), A049778.

A095800 := proc(n,k) k/4*( (2*n+1)*(-1)^(n+k)+2*k-1) ; abs(%) ; end proc:
seq(seq(A095800(n,k),k=1..n),n=1..16) ; # R. J. Mathar, Apr 17 2011

T(n,k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 );
for(n=1,20,for(m=1,n,print1(T(n,m),", ")));
\\ Joerg Arndt, Mar 05 2014

# Generates the b-file
i=1
for n in range(1,126):
    for k in range(1,n+1):
        print(str(i)+" "+str(abs(k*((2*n+1)*(-1)**(n+k)+2*k-1)//4)))
        i+=1 # Indranil Ghosh, Feb 17 2017

Let M(n,k) = (-1)^(k+1)*k, 1<=k<=n be the infinite lower triangular matrix with 1, -2, 3,.. up to the diagonal, and the upper triangular part all zeros. The 3x3 submatrix would be [1 0 0 / 1 -2 0 / 1 -2 3]. The current triangle contains the absolute values of the matrix square M^2.

Replaced NAME by closed form and inserted a missing row. - R. J. Mathar, Apr 17 2011

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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A213849 Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.

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A095800 Triangle T(n,k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 ) read by rows, 1<=k<=n.

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A134447 A002260 * A128174.

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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.

A213849 Rectangular array: (row n) = bc, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and = convolution.

A095800 Triangle T(n,k) = abs( k ( (2n+1)(-1)^(n+k)+2k-1) /4 ) read by rows, 1<=k<=n.