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%I #38 Mar 23 2020 09:43:02
%S 1,1,1,1,2,1,1,4,3,1,1,7,7,4,1,1,11,13,10,5,1,1,16,21,19,13,6,1,1,22,
%T 31,31,25,16,7,1,1,29,43,46,41,31,19,8,1,1,37,57,64,61,51,37,22,9,1,1,
%U 46,73,85,85,76,61,43,25,10,1,1,56,91,109,113,106,91,71,49,28,11,1,1,67
%N Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.
%C Row n gives the centered figurate numbers of the n-gon.
%C Antidiagonal sums are in A101338.
%F T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
%F T(10,m) = A062786(m+1).
%F T(11,m) = A069125(m+1).
%F T(12,m) = A003154(m+1).
%F T(13,m) = A069126(m+1).
%F T(14,m) = A069127(m+1).
%F T(15,m) = A069128(m+1).
%F T(16,m) = A069129(m+1).
%F T(17,m) = A069130(m+1).
%F T(18,m) = A069131(m+1).
%F T(19,m) = A069132(m+1).
%F T(20,m) = A069133(m+1).
%F T(n+1,m) = T(n,m) + m*(m+1)/2. - _Gary W. Adamson_ and _Michel Marcus_, Oct 13 2015
%e The upper left corner of the infinite array T is
%e |0| 1 1 1 1 1 1 1 1 1 1 ... A000012
%e |1| 1 2 4 7 11 16 22 29 37 46 ... A000124
%e |2| 1 3 7 13 21 31 43 57 73 91 ... A002061
%e |3| 1 4 10 19 31 46 64 85 109 136 ... A005448
%e |4| 1 5 13 25 41 61 85 113 145 181 ... A001844
%e |5| 1 6 16 31 51 76 106 141 181 226 ... A005891
%e |6| 1 7 19 37 61 91 127 169 217 271 ... A003215
%e |7| 1 8 22 43 71 106 148 197 253 316 ... A069099
%e |8| 1 9 25 49 81 121 169 225 289 361 ... A016754
%e |9| 1 10 28 55 91 136 190 253 325 406 ... A060544
%p A101321 := proc(n,k)
%p n*k*(k+1)/2+1 ;
%p end proc: # _R. J. Mathar_, Jul 28 2016
%t T[n_, m_] := 1 + n m (m + 1)/2;
%t Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 23 2020 *)
%o (Iverson's J language) Let cfn be the formula above. Then the first 20 rows and columns of T are: T =: cfn / ~ i. 20 where i.
%o (PARI) T(n,m) = 1 + n*m*(m+1)/2 \\ _Charles R Greathouse IV_, Jul 28 2016
%Y Cf. A000217, A001263, A101338.
%K easy,nonn,tabl
%O 0,5
%A Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004
%E Edited by _R. J. Mathar_, Oct 21 2009
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