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A164844 Generalized Pascal Triangle - satisfying the same recurrence as Pascal's triangle, but with a(n,0)=1 and a(n,n)=10^n (instead of both being 1).

%I A164844 #38 Oct 07 2024 06:32:23
%S A164844 1,1,10,1,11,100,1,12,111,1000,1,13,123,1111,10000,1,14,136,1234,
%T A164844 11111,100000,1,15,150,1370,12345,111111,1000000,1,16,165,1520,13715,
%U A164844 123456,1111111,10000000,1,17,181,1685,15235,137171,1234567,11111111,100000000,1,18,198,1866,16920,152406,1371738,12345678,111111111,1000000000,1,19,216,2064,18786,169326,1524144,13717416,123456789,1111111111,10000000000
%N A164844 Generalized Pascal Triangle - satisfying the same recurrence as Pascal's triangle, but with a(n,0)=1 and a(n,n)=10^n (instead of both being 1).
%C A164844 Like with Pascal's triangle, the columns grown polynomially. For example, a(n,1)=10+n, a(n,2)=(1/2)*(180+19n+n^2), a(n,3)=(1/6)*(5400 + 569n + 30n^2 + n^3). Likewise, diagonals grow exponentially: a(n,n)=10^n, a(n,n-1) = (10^n-1) / 9. - _Kellen Myers_, Jan 24 2010
%H A164844 Robert Israel, <a href="/A164844/b164844.txt">Table of n, a(n) for n = 0..10152</a> (rows 0 to 141, flattened)
%F A164844 From  _Kellen Myers_, Jan 24 2010: (Start)
%F A164844 a(n,k) = Sum_{i = 0..k} 10^i * binomial(n-i-1, n-k-1), for 0<=k<=n.
%F A164844 a(n,0) = 1, a(n,n) = 10^n, a(n,k) = a(n-1,k-1)+a(n-1,k). (End)
%F A164844 T(n,k) = T(n-1,k)+11*T(n-1,k-1)-10*T(n-2,k-1)-10*T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=10, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Dec 27 2013
%F A164844 G.f. of triangle: g(x,y) = (1-xy)/((1-10xy)(1-x-xy)). - _Robert Israel_, Jul 01 2016
%e A164844 Triangle begins:
%e A164844   1
%e A164844   1,10
%e A164844   1,11,100
%e A164844   1,12,111,1000
%e A164844   1,13,123,1111,10000
%e A164844   1,14,136,1234,11111,100000
%p A164844 f:= proc(n,k) option remember;
%p A164844 if k=n then 10^n elif k=0 then 1 else procname(n-1,k-1)+procname(n-1,k) fi
%p A164844 end proc:
%p A164844 seq(seq(f(n,k),k=0..n),n=0..10); # _Robert Israel_, Jul 01 2016
%t A164844 f[r_, k_] := Sum[10^i*Binomial[r - i - 1, r - k - 1], {i, 0, k}]; TableForm[Table[f[n, k], {n, 0, 15}, {k, 0, n}]] (* _Alex Meiburg_, Aug 21 2010 *)
%t A164844 a[n_, k_] := a[n, k] = Piecewise[{{0, k > n || k < 0}, {1, k == 0}, {10^n, k == n}}, a[n - 1, k - 1] + a[n - 1, k]]; TableForm[Table[a[n, k], {n, 0, 10}, {k, 0, n}]] (* _Kellen Myers_, Jan 24 2010 *)
%Y A164844 Cf. A007318, A093645, A011557, A228196.
%K A164844 nonn,tabl,easy
%O A164844 0,3
%A A164844 _Mark Dols_, Aug 28 2009
%E A164844 Definition clarified, more terms, and revision of Meiburg's Mathematica code by _Kellen Myers_, Jan 24 2010