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%I #21 Aug 15 2022 04:23:53
%S 1,3,4,7,13,21,15,40,85,156,31,121,341,781,1555,63,364,1365,3906,9331,
%T 19608,127,1093,5461,19531,55987,137257,299593,255,3280,21845,97656,
%U 335923,960800,2396745,5380840,511,9841,87381,488281,2015539,6725601,19173961,48427561,111111111
%N Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.
%H G. C. Greubel, <a href="/A125118/b125118.txt">Rows n = 1..50 of the triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%F T(n, k) = Sum_{i=0..n-1} (k+1)^i.
%F T(n+1, k) = (k+1)*T(n, k) + 1.
%F Sum_{k=1..n} T(n, k) = A125120(n).
%F T(2*n-1, n) = A125119(n).
%F T(n, 1) = A000225(n).
%F T(n, 2) = A003462(n) for n>1.
%F T(n, 3) = A002450(n) for n>2.
%F T(n, 4) = A003463(n) for n>3.
%F T(n, 5) = A003464(n) for n>4.
%F T(n, 9) = A002275(n) for n>8.
%F T(n, n) = A060072(n+1).
%F T(n, n-1) = A023037(n) for n>1.
%F T(n, n-2) = A031973(n) for n>2.
%F T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - _Mathew Englander_, Dec 19 2020
%e First 4 rows:
%e 1: [1]_2
%e 2: [11]_2 ........ [11]_3
%e 3: [111]_2 ....... [111]_3 ....... [111]_4
%e 4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
%e _
%e 1: 1
%e 2: 2+1 ........... 3+1
%e 3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
%e 4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.
%t Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Aug 15 2022 *)
%o (Magma) [((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // _G. C. Greubel_, Aug 15 2022
%o (SageMath)
%o def A125118(n,k): return ((k+1)^n -1)/k
%o flatten([[A125118(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Aug 15 2022
%Y This triangle shares some features with triangle A104878.
%Y This triangle is a portion of rectangle A055129.
%Y Each term of A110737 comes from the corresponding row of this triangle.
%Y Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.
%Y Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
%Y Cf. A023037, A031973, A125119, A125120 (row sums).
%K nonn,tabl,base
%O 1,2
%A _Reinhard Zumkeller_, Nov 21 2006
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