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A227076 A triangle formed like Pascal's triangle, but with 5^n on the borders instead of 1.

%I A227076 #18 Jan 11 2025 03:54:16
%S A227076 1,5,5,25,10,25,125,35,35,125,625,160,70,160,625,3125,785,230,230,785,
%T A227076 3125,15625,3910,1015,460,1015,3910,15625,78125,19535,4925,1475,1475,
%U A227076 4925,19535,78125,390625,97660,24460,6400,2950,6400,24460,97660,390625
%N A227076 A triangle formed like Pascal's triangle, but with 5^n on the borders instead of 1.
%C A227076 All rows except the zeroth are divisible by 5. Is there a closed-form formula for these numbers, as there is for binomial coefficients?
%H A227076 T. D. Noe, <a href="/A227076/b227076.txt">Rows n = 0..50 of triangle, flattened</a>
%F A227076 From _R. J. Mathar_, Aug 09 2013: (Start)
%F A227076 T(n,0) = 5^n.
%F A227076 T(n,1) = 5*A047850(n-1).
%F A227076 T(n,2) = 5*(5^n/80 + 3*n/4 + 51/16).
%F A227076 T(n,3) = 5*(5^n/320 + 45*n/16 + 3*n^2/8 + 819/64). (End)
%F A227076 Sum_{k=0..n} (-1)^k*T(n, k) = 20*(1+(-1)^n)*A009969(floor((n-1)/2)) - (3/5)*[n = 0]. - _G. C. Greubel_, Jan 10 2025
%e A227076 Triangle begins as:
%e A227076        1;
%e A227076        5,     5;
%e A227076       25,    10,    25;
%e A227076      125,    35,    35,  125;
%e A227076      625,   160,    70,  160,  625;
%e A227076     3125,   785,   230,  230,  785, 3125;
%e A227076    15625,  3910,  1015,  460, 1015, 3910, 15625;
%e A227076    78125, 19535,  4925, 1475, 1475, 4925, 19535, 78125;
%e A227076   390625, 97660, 24460, 6400, 2950, 6400, 24460, 97660, 390625;
%p A227076 A227076 := proc(n,k)
%p A227076     if k = 0 or k = n then
%p A227076         5^n ;
%p A227076     elif k < 0 or k > n then
%p A227076         0;
%p A227076     else
%p A227076         procname(n-1,k)+procname(n-1,k-1) ;
%p A227076     end if;
%p A227076 end proc: # _R. J. Mathar_, Aug 09 2013
%t A227076 t = {}; Do[r = {}; Do[If[k == 0 || k == n, m = 5^n, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]
%o A227076 (Magma)
%o A227076 function T(n,k) // T = A227076
%o A227076   if k eq 0 or k eq n then return 5^n;
%o A227076   else return T(n-1,k) + T(n-1,k-1);
%o A227076   end if;
%o A227076 end function;
%o A227076 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 10 2025
%o A227076 (Python)
%o A227076 from sage.all import *
%o A227076 @CachedFunction
%o A227076 def T(n,k): # T = A227076
%o A227076     if k==0 or k==n: return pow(5,n)
%o A227076     else: return T(n-1,k) + T(n-1,k-1)
%o A227076 print(flatten([[T(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 10 2025
%Y A227076 Cf. A007318 (Pascal's triangle), A228053 ((-1)^n on the borders).
%Y A227076 Cf. A051601 (n on the borders), A137688 (2^n on borders).
%Y A227076 Cf. A083585 (row sums: (8*5^n - 5*2^n)/3), A227074 (4^n edges), A227075 (3^n edges).
%Y A227076 Cf. A000351.
%K A227076 nonn,tabl
%O A227076 0,2
%A A227076 _T. D. Noe_, Aug 06 2013