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A130424 Main diagonal of array A[k,n] = n-th sum of k consecutive k-gonal numbers, k>2.

%I A130424 #19 Feb 16 2025 08:33:06
%S A130424 4,30,125,365,854,1724,3135,5275,8360,12634,18369,25865,35450,47480,
%T A130424 62339,80439,102220,128150,158725,194469,235934,283700,338375,400595,
%U A130424 471024,550354,639305,738625,849090,971504,1106699,1255535,1418900
%N A130424 Main diagonal of array A[k,n] = n-th sum of k consecutive k-gonal numbers, k>2.
%C A130424 The first row of the array is the sum of 3 consecutive triangular numbers = A000217(n) + A000217(n+1) + A000217(n+2) = Centered triangular numbers: 3*n*(n-1)/2 + 1, for n>1. The second row of the array is the sum of 4 consecutive squares = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 = A027575(n). The third row of the array is the sum of 5 consecutive pentagonal numbers.
%H A130424 Harvey P. Dale, <a href="/A130424/b130424.txt">Table of n, a(n) for n = 1..1000</a>
%H A130424 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.
%H A130424 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A130424 a(n) = A[n+2,n] = P(k+2,n) + P(k+2,n+1) + P(k+2,n+2) + ... P(k+2,n+k-1) where P(k,n) = k*((n-2)*k - (n-4))/2.
%F A130424 a(n) = (n+2)*(7*n^3-8*n^2+12*n-3)/6. [_R. J. Mathar_, Oct 30 2008]
%F A130424 G.f.: x*(4+10*x+15*x^2-x^4)/(1-x)^5. [_Colin Barker_, Sep 08 2012]
%e A130424 The array begins:
%e A130424 k / A[k,n]
%e A130424 3.|...4..10..19...31...46...64...85..109.136.166...=A005448(n+1).
%e A130424 4.|..14..30..54...86..126..174..230..294.366.446...=A027575(n).
%e A130424 5.|..40..75.125..190..270..365..475..600.740...
%e A130424 6.|..95.161.251..365..503..665..851.1061.1295...
%e A130424 7.|.196.308.455..637..854.1106.1393.1715.2072...
%e A130424 8.|.364.540.764.1036.1356.1724.2140.2604.3116...
%p A130424 P := proc(k,n) n*((k-2)*n-k+4)/2 ; end: A := proc(k,n) add( P(k,i),i=n..n+k-1) ; end: A130424 := proc(n) A(n+3,n) ; end: seq(A130424(n),n=0..40) ; # _R. J. Mathar_, Oct 28 2007
%t A130424 LinearRecurrence[{5,-10,10,-5,1},{4,30,125,365,854},50] (* _Harvey P. Dale_, Jun 23 2020 *)
%Y A130424 Cf. A000217, A000290, A000326, A000384, A000566, A000567, A005448, A005918, A016825, A017377, A129803, A129863, A130423.
%K A130424 easy,nonn
%O A130424 1,1
%A A130424 _Jonathan Vos Post_, May 26 2007
%E A130424 More terms from _R. J. Mathar_, Oct 28 2007