The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution. 5
1, 5, 3, 16, 13, 7, 42, 38, 29, 15, 99, 94, 82, 61, 31, 219, 213, 198, 170, 125, 63, 466, 459, 441, 406, 346, 253, 127, 968, 960, 939, 897, 822, 698, 509, 255, 1981, 1972, 1948, 1899, 1809, 1654, 1402, 1021, 511, 4017, 4007, 3980, 3924, 3819, 3633 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Principal diagonal: A213572.
Antidiagonal sums: A213581.
Row 1, (1,2,3,4,5,...)**(1,3,7,15,31,...): A002662.
Row 2, (1,2,3,4,5,...)**(3,7,15,31,63,...).
Row 3, (1,2,3,4,5,...)**(7,15,31,63,...).
For a guide to related arrays, see A213500.
LINKS
FORMULA
T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.
T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - G. C. Greubel, Jul 25 2019
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1, 5, 16, 42, 99, 219, ...
3, 13, 38, 94, 213, 459, ...
7, 29, 82, 198, 441, 939, ...
15, 61, 170, 406, 897, 1899, ...
31, 125, 346, 822, 1809, 3819, ...
...
MATHEMATICA
(* First program *)
b[n_]:= n; c[n_]:= -1 + 2^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}] (* A213571 *)
d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
(* Additional programs *)
Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)
PROG
(PARI) for(n=1, 12, for(k=1, n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Jul 25 2019
(Magma) [2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019
(Sage) [[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # G. C. Greubel, Jul 25 2019
CROSSREFS
Sequence in context: A053371 A199005 A217415 * A185880 A292243 A105201
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Jun 19 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 20 08:57 EDT 2024. Contains 372710 sequences. (Running on oeis4.)