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).

1, 1, 10, 1, 11, 100, 1, 12, 111, 1000, 1, 13, 123, 1111, 10000, 1, 14, 136, 1234, 11111, 100000, 1, 15, 150, 1370, 12345, 111111, 1000000, 1, 16, 165, 1520, 13715, 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

Offset

0,3

Comments

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]

Links

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened)

Formula

From Kellen Myers, Jan 24 2010: (Start)

a(n,k) = Sum_{i = 0..k} 10^i * binomial(n-i-1, n-k-1)), for 0<=k<=n.

a(n,0) = 1, a(n,n) = 10^n, a(n,k) = a(n-1,k-1)+a(n-1,k). (End)

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

G.f. of triangle: g(x,y) = (1-xy)/((1-10xy)(1-x-xy)). - Robert Israel, Jul 01 2016

Example

Triangle begins:
1
1,10
1,11,100
1,12,111,1000
1,13,123,1111,10000
1,14,136,1234,11111,100000

Maple

f:= proc(n, k) option remember;
if k=n then 10^n elif k=0 then 1 else procname(n-1, k-1)+procname(n-1, k) fi
end proc:
seq(seq(f(n, k), k=0..n), n=0..10); # Robert Israel, Jul 01 2016

Mathematica

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 *)
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 *)

Crossrefs

Cf. A007318, A093645, A011557, A228196.

Sequence in context: A172171 A327723 A164899 * A287015 A130311 A063672

Adjacent sequences: A164841 A164842 A164843 * A164845 A164846 A164847

Keyword

nonn,tabl

Author

Mark Dols, Aug 28 2009

Extensions

Definition clarified, more terms, and revision of Meiburg's Mathematica code by Kellen Myers, Jan 24 2010

Status

approved