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Table T(m,k)=m^k-k^m (with 0^0 taken to be 1) as square array read by antidiagonals.
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7
0, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 2, 0, -2, -1, 1, 3, 1, -1, -3, -1, 1, 4, 0, 0, 0, -4, -1, 1, 5, -7, -17, 17, 7, -5, -1, 1, 6, -28, -118, 0, 118, 28, -6, -1, 1, 7, -79, -513, -399, 399, 513, 79, -7, -1, 1, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, -1, 1, 9, -431
Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.
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4
1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625
FORMULA
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)
EXAMPLE
A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1 0 0 0 0 0 0 0 0 0 0 ...
1: 0 1 2 3 4 5 6 7 8 9 10 ...
2: 0 2 16 72 256 800 2304 6272 16384 41472 102400 ...
3: 0 3 72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 0 0
2: 0 1 0
3: 0 2 2 0
4: 0 3 16 3 0
5: 0 4 72 72 4 0
6: 0 5 256 729 256 5 0
7: 0 6 800 5184 5184 800 6 0
8: 0 7 2304 30375 65536 30375 2304 7 0
9: 0 8 6272 157464 640000 640000 157464 6272 8 0
MATHEMATICA
{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)
PROG
(PARI) t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018
Triangle read by rows: T(n,r) = n^r + r^n (1 <= r <= n).
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2
2, 3, 8, 4, 17, 54, 5, 32, 145, 512, 6, 57, 368, 1649, 6250, 7, 100, 945, 5392, 23401, 93312, 8, 177, 2530, 18785, 94932, 397585, 1647086, 9, 320, 7073, 69632, 423393, 1941760, 7861953, 33554432, 10, 593, 20412, 268705, 2012174, 10609137, 45136576
EXAMPLE
2,
3, 8,
4, 17, 54,
5, 32, 145, 512,
6, 57, 368, 1649, 6250,
7, 100, 945, 5392, 23401, 93312,
MAPLE
T:=(n, r)->n^r+r^n: for n from 1 to 10 do seq(T(n, r), r=1..n) od; # yields sequence in triangular form # Emeric Deutsch, Feb 04 2006
MATHEMATICA
Flatten[Table[n^r+r^n, {n, 10}, {r, n}]] (* Harvey P. Dale, Jun 19 2011 *)
Concatenate n and the sum of primes dividing n (counting multiplicity).
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2
10, 22, 33, 44, 55, 65, 77, 86, 96, 107, 1111, 127, 1313, 149, 158, 168, 1717, 188, 1919, 209, 2110, 2213, 2323, 249, 2510, 2615, 279, 2811, 2929, 3010, 3131, 3210, 3314, 3419, 3512, 3610, 3737, 3821, 3916, 4011, 4141, 4212, 4343, 4415, 4511, 4625, 4747
MAPLE
f:= proc(n) local q;
q:= add(t[1]*t[2], t=ifactors(n)[2]);
10^(1+ilog10(q))*n+q
end proc:
f(1):= 10:
MATHEMATICA
pr[{a_, b_}]:=a*b; Join[{10}, Table[FromDigits[Flatten[IntegerDigits[Join[{n}, {Total[pr/@FactorInteger[n]]}]]]], {n, 2, 47}]] (* James C. McMahon, Apr 02 2024 *)
Triangle T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1, read by rows.
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2
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1
COMMENTS
This sequence is an approximation of Pascal's triangle with interior Kurtosis.
FORMULA
T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1.
T(n, k) = T(n, n-k).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 8, 4, 1;
1, 5, 17, 17, 5, 1;
1, 6, 32, 54, 32, 6, 1;
1, 7, 57, 145, 145, 57, 7, 1;
1, 8, 100, 368, 512, 368, 100, 8, 1;
1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1;
1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1;
1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1;
The interior Kurtosis, T(n,k) - binomial(n, k), is:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 2, 0, 0;
0, 0, 7, 7, 0, 0;
0, 0, 17, 34, 17, 0, 0;
0, 0, 36, 110, 110, 36, 0, 0;
0, 0, 72, 312, 442, 312, 72, 0, 0;
0, 0, 141, 861, 1523, 1523, 861, 141, 0, 0;
0, 0, 275, 2410, 5182, 5998, 5182, 2410, 275, 0, 0;
0, 0, 538, 6908, 18455, 22939, 22939, 18455, 6908, 538, 0, 0;
MATHEMATICA
T[n_, k_]:= If[n==0, 1, (k^(n-k) + (n-k)^k)];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Sage) flatten([[1 if k==n else k^(n-k) + (n-k)^k for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 07 2021 (Magma) [k eq 0 select 1 else k^(n-k) + (n-k)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2021
a(1) = 1; a(n+1) = a(n)^n + n^a(n).
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0
COMMENTS
Next term is too large to display: 2.3459495195697547514*10^4258
MATHEMATICA
a=1; lst={}; Do[a=a^n+n^a; AppendTo[lst, IntegerPart[a]], {n, 0, 4}]; lst
nxt[{n_, a_}] := {n + 1, a^n + n^a}; NestList[nxt, {1, 1}, 4][[All, 2]] (* Harvey P. Dale, Jul 18 2021 *)
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