A173075 -id:A173075

Search: a173075 -id:a173075

Displaying 1-6 of 6 results found.

page 1

A132044

Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows.

+10
15

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 14, 19, 14, 5, 1, 1, 6, 20, 34, 34, 20, 6, 1, 1, 7, 27, 55, 69, 55, 27, 7, 1, 1, 8, 35, 83, 125, 125, 83, 35, 8, 1, 1, 9, 44, 119, 209, 251, 209, 119, 44, 9, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...).

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 12 2021

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

T(n, k) = A007318(n,k) + A103451(n,k) - A000012(n,k), an infinite lower triangular matrix.

T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - Roger L. Bagula, Feb 08 2010

From G. C. Greubel, Feb 12 2021: (Start)

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 0.

Sum_{k=0..n} T(n, k, 0) = 2^n - (n-1) - [n=0]. (End)

EXAMPLE

First few rows of the triangle:

1, 1;

1, 1, 1;

1, 2, 2, 1;

1, 3, 5, 3, 1;

1, 4, 9, 9, 4, 1;

1, 5, 14, 19, 14, 5, 1;

1, 6, 20, 34, 34, 20, 6, 1;

1, 7, 27, 55, 69, 55, 27, 7, 1;

MATHEMATICA

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 08 2010 *)

PROG

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1

flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >;

[T(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021

CROSSREFS

Cf. A007318, A103451, A132045.

Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3).

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 08 2007

STATUS

approved

A167763

Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

+10
6

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 7, 4, 1, 0, 1, 5, 12, 12, 5, 1, 0, 1, 6, 18, 30, 18, 6, 1, 0, 1, 7, 25, 55, 55, 25, 7, 1, 0, 1, 8, 33, 88, 143, 88, 33, 8, 1, 0, 1, 9, 42, 130, 273, 273, 130, 42, 9, 1, 0, 1, 10, 52, 182, 455, 728, 455, 182, 52, 10, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

See A118340 for definition of pendular triangles and pendular sums.

The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - Tom Copeland, Jan 22 2020

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

FORMULA

T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.

If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - Paul D. Hanna, Nov 12 2009

Sum_{k=0..n} T(n, k, p=0) = A093951(n). - G. C. Greubel, Feb 17 2021

EXAMPLE

Triangle begins:

1, 0;

1, 1, 0;

1, 2, 1, 0;

1, 3, 3, 1, 0;

1, 4, 7, 4, 1, 0;

1, 5, 12, 12, 5, 1, 0; ...

MATHEMATICA

T[n_, k_, p_]:= T[n, k, p] = If[n<k || k<0, 0, If[k==0, 1, If[k==n, 0, If[n<=2*k, T[n, n-k-1, p] + p*T[n-1, k, p], T[n, n-k, p] + T[n-1, k, p] ]]]];

Table[T[n, k, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

PROG

(PARI) {T(n, k)=if(k==0, 1, if(n==k, 0, if(n>2*k, binomial(n+k+1, k)*(n-2*k+1)/(n+k+1), T(n, n-1-k))))} \\ Paul D. Hanna, Nov 12 2009

(Sage)

@CachedFunction

def T(n, k, p):

if (k<0 or n<k): return 0

elif (k==0): return 1

elif (k==n): return 0

elif (n>2*k): return T(n, n-k, p) + T(n-1, k, p)

else: return T(n, n-k-1, p) + p*T(n-1, k, p)

flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

(Magma)

function T(n, k, p)

if k lt 0 or n lt k then return 0;

elif k eq 0 then return 1;

elif k eq n then return 0;

elif n gt 2*k then return T(n, n-k, p) + T(n-1, k, p);

else return T(n, n-k-1, p) + p*T(n-1, k, p);

end if;

return T;

end function;

[T(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

CROSSREFS

Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).

Cf. A001764, A006013, A006629, A093951, A102893, A173075.

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Nov 11 2009

STATUS

approved

A173046

Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2, read by rows.

+10
4

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 37, 19, 1, 1, 36, 105, 105, 36, 1, 1, 69, 270, 403, 270, 69, 1, 1, 134, 660, 1314, 1314, 660, 134, 1, 1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1, 1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1, 1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2.

Sum_{k=0..n} T(n, k, 2) = 4^(n-1) + 2^n - (n-1) - (5/4)*[n=0] = A000302(n-1) + A132045(n) - (5/4)*[n=0]. - [n=1]. - G. C. Greubel, Feb 16 2021

EXAMPLE

Triangle begins as:

1, 1;

1, 5, 1;

1, 10, 10, 1;

1, 19, 37, 19, 1;

1, 36, 105, 105, 36, 1;

1, 69, 270, 403, 270, 69, 1;

1, 134, 660, 1314, 1314, 660, 134, 1;

1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1;

1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1;

1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1;

MATHEMATICA

T[n_, m_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

PROG

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1

flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >;

[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

CROSSREFS

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).

Cf. A000302, A132045.

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 08 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 16 2021

STATUS

approved

A173047

Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3, read by rows.

+10
4

1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 167, 84, 1, 1, 247, 738, 738, 247, 1, 1, 734, 2930, 4393, 2930, 734, 1, 1, 2193, 10955, 21904, 21904, 10955, 2193, 1, 1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1, 1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3.

Sum_{k=0..n} T(n, k, 3) = (1/4)*(6^n + 2^(n+2) - 4*(n-1) - 5*[n=0] - 6*[n=1]). - G. C. Greubel, Feb 16 2021

EXAMPLE

Ttiangle begins as:

1, 1;

1, 10, 1;

1, 29, 29, 1;

1, 84, 167, 84, 1;

1, 247, 738, 738, 247, 1;

1, 734, 2930, 4393, 2930, 734, 1;

1, 2193, 10955, 21904, 21904, 10955, 2193, 1;

1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1;

1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1;

MATHEMATICA

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

PROG

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1

flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >;

[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

CROSSREFS

Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (q=3).

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 08 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 16 2021

STATUS

approved

A173076

Triangle T(n, k, q) = binomial(n, k) - 1 + q^(floor(n/2))*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 2, read by rows.

+10
3

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 7, 13, 7, 1, 1, 8, 21, 21, 8, 1, 1, 13, 46, 67, 46, 13, 1, 1, 14, 60, 114, 114, 60, 14, 1, 1, 23, 123, 295, 389, 295, 123, 23, 1, 1, 24, 147, 419, 685, 685, 419, 147, 24, 1, 1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 2.

EXAMPLE

Triangle begins as:

1, 1;

1, 3, 1;

1, 4, 4, 1;

1, 7, 13, 7, 1;

1, 8, 21, 21, 8, 1;

1, 13, 46, 67, 46, 13, 1;

1, 14, 60, 114, 114, 60, 14, 1;

1, 23, 123, 295, 389, 295, 123, 23, 1;

1, 24, 147, 419, 685, 685, 419, 147, 24, 1;

1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1;

MATHEMATICA

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(Floor[n/2])*Binomial[n-2, k-1]];

Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^(Floor(n/2))*Binomial(n-2, k-1) -1 >;

[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^(n//2)*binomial(n-2, k-1) -1

flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)])

CROSSREFS

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173077 (q=3).

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 09 2010

EXTENSIONS

Edited by G. C. Greubel, Jul 09 2021

STATUS

approved

A173077

Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

+10
3

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 12, 23, 12, 1, 1, 13, 36, 36, 13, 1, 1, 32, 122, 181, 122, 32, 1, 1, 33, 155, 304, 304, 155, 33, 1, 1, 88, 513, 1270, 1689, 1270, 513, 88, 1, 1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1, 1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.

EXAMPLE

Triangle starts:

1, 1;

1, 4, 1;

1, 5, 5, 1;

1, 12, 23, 12, 1;

1, 13, 36, 36, 13, 1;

1, 32, 122, 181, 122, 32, 1;

1, 33, 155, 304, 304, 155, 33, 1;

1, 88, 513, 1270, 1689, 1270, 513, 88, 1;

1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1;

1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;

...

Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...

MATHEMATICA

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^(Floor(n/2))*Binomial(n-2, k-1) -1 >;

[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^(n//2)*binomial(n-2, k-1) -1

flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

CROSSREFS

Cf. A132044 (q=0), A173075 (q=1), A173076 (q=2), this sequence (q=3).

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 09 2010

STATUS

approved

page 1

Search completed in 0.007 seconds