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%I #41 Apr 26 2020 15:18:29
%S 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,
%T 34,19,7,120,112,98,80,60,40,22,8,165,156,140,119,95,70,46,25,9,220,
%U 210,192,168,140,110,80,52,28,10,286,275,255,228,196,161,125,90
%N Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.
%C Principal diagonal: A002412.
%C Antidiagonal sums: A002415.
%C Row 1: (1,2,3,...)**(1,2,3,...) = A000292.
%C Row 2: (1,2,3,...)**(2,3,4,...) = A005581.
%C Row 3: (1,2,3,...)**(3,4,5,...) = A006503.
%C Row 4: (1,2,3,...)**(4,5,6,...) = A060488.
%C Row 5: (1,2,3,...)**(5,6,7,...) = A096941.
%C Row 6: (1,2,3,...)**(6,7,8,...) = A096957.
%C ...
%C 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.
%C ...
%C 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.
%C b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
%C h .......... h .......... A213500 . A002412 . A002415
%C h .......... h^2 ........ A212891 . A213436 . A024166
%C h^2 ........ h .......... A213503 . A117066 . A033455
%C h^2 ........ h^2 ........ A213505 . A213546 . A213547
%C h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
%C h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
%C h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
%C h .......... h^3 ........ A213553 . A213554 . A101089
%C h^3 ........ h .......... A213555 . A213556 . A213547
%C h^3 ........ h^3 ........ A213558 . A213559 . A213560
%C h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
%C h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
%C 2^(h-1) .... h .......... A213568 . A213569 . A047520
%C 2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
%C h .......... Fibo(h) .... A213576 . A213577 . A213578
%C Fibo(h) .... h .......... A213579 . A213580 . A053808
%C Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
%C Fibo(h+1) .. h .......... A213584 . A213585 . A213586
%C Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
%C h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
%C Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
%C h .......... -1+2^h ..... A213571 . A213572 . A213581
%C -1+2^h ..... h .......... A213582 . A213583 . A156928
%C -1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
%C h .......... 2*h-1 ...... A213750 . A007585 . A002417
%C 2*h-1 ...... h .......... A213751 . A051662 . A006325
%C 2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
%C 2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
%C -1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
%C 2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
%C 2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
%C Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
%C Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
%C Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
%C h .......... 1+[h/2] .... A213778 . A213779 . A213780
%C 1+[h/2] .... h .......... A213781 . A213782 . A005712
%C 1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
%C h .......... 3*h-2 ...... A213761 . A172073 . A002419
%C 3*h-2 ...... h .......... A213771 . A213772 . A132117
%C 3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
%C h .......... 3*h-1 ...... A213819 . A213820 . A153978
%C 3*h-1 ...... h .......... A213821 . A033431 . A176060
%C 3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
%C 3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
%C 3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
%C 2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
%C 3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
%C h .......... 4*h-3 ...... A213835 . A172078 . A051797
%C 4*h-3 ...... h .......... A213836 . A213837 . A071238
%C 4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
%C 2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
%C 2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
%C 4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
%C [(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
%C h .......... C(2*h-2,h-1) A213853
%C ...
%C 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.
%H Clark Kimberling, <a href="/A213500/b213500.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
%F T(n,k) = 2*T(n-1,k) - T(n-2,k).
%F G.f. for row n: x*(n - (n - 1)*x)/(1 - x)^4.
%e Northwest corner (the array is read by southwest falling antidiagonals):
%e 1, 4, 10, 20, 35, 56, 84, ...
%e 2, 7, 16, 30, 50, 77, 112, ...
%e 3, 10, 22, 40, 65, 98, 140, ...
%e 4, 13, 28, 50, 80, 119, 168, ...
%e 5, 16, 34, 60, 95, 140, 196, ...
%e 6, 19, 40, 70, 110, 161, 224, ...
%e T(6,1) = (1)**(6) = 6;
%e T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
%e T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.
%t b[n_] := n; c[n_] := n
%t t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t r[n_] := Table[t[n, k], {k, 1, 60}] (* A213500 *)
%o (PARI)
%o t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
%o tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
%o tabl(12) \\ _Indranil Ghosh_, Mar 26 2017
%o (Python)
%o def t(n, k): return sum((k - i) * (n + i) for i in range(k))
%o for n in range(1, 13):
%o print([t(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 26 2017
%Y Cf. A000027.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Jun 14 2012
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