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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). 9
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 (list; table; graph; refs; listen; history; text; internal format)
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: A297352 A172171 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

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Last modified June 25 23:52 EDT 2019. Contains 324367 sequences. (Running on oeis4.)