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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. 89
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Clark Kimberling, Antidiagonals n = 1..60, flattened FORMULA 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. EXAMPLE 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. MATHEMATICA 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 *) PROG (PARI) 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 (Python) def t(n, k): return sum([(k - i) * (n + i) for i in xrange(0, k)]) for n in xrange(1, 13): ....print [t(k, n - k + 1) for k in xrange(1, n + 1)] # Indranil Ghosh, Mar 26 2017 CROSSREFS Cf. A000027. Sequence in context: A266418 A193422 A160572 * A213584 A213587 A066579 Adjacent sequences:  A213497 A213498 A213499 * A213501 A213502 A213503 KEYWORD nonn,easy,tabl AUTHOR Clark Kimberling, Jun 14 2012 STATUS approved

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