A131111 -id:A131111

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A131115

Triangle read by rows: T(n,k) = 7*binomial(n,k) for 1 <= k <= n with T(n,n) = 1 for n >= 0.

+10
7

1, 7, 1, 7, 14, 1, 7, 21, 21, 1, 7, 28, 42, 28, 1, 7, 35, 70, 70, 35, 1, 7, 42, 105, 140, 105, 42, 1, 7, 49, 147, 245, 245, 147, 49, 1, 7, 56, 196, 392, 490, 392, 196, 56, 1, 7, 63, 252, 588, 882, 882, 588, 252, 63, 1, 7, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

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

OFFSET

0,2

COMMENTS

Row sums give A048489.

Non-diagonal entries of Pascal's triangle are multiplied by 7. - Emeric Deutsch, Jun 20 2007

The matrix inverse starts

-7, 1;

91, -14, 1;

-1771, 273, -21, 1;

45955, -7084, 546, -28, 1;

-1490587, 229775, -17710, 910, -35, 1;

58018051, -8943522, 689325, -35420, 1365, -42, 1;

-2634606331, 406126357, -31302327, 1608425, -61985, 1911, -49, 1;

... - R. J. Mathar, Mar 15 2013

LINKS

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

FORMULA

G.f.: (1 + 6*x - t*x)/((1-t*x)*(1-x-t*x)). - Emeric Deutsch, Jun 20 2007

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

7, 1;

7, 14, 1;

7, 21, 21, 1;

7, 28, 42, 28, 1;

7, 35, 70, 70, 35, 1;

...

MAPLE

T := proc (n, k) if k < n then 7*binomial(n, k) elif k = n then 1 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 20 2007

MATHEMATICA

Table[If[k==n, 1, 7*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PROG

(PARI) T(n, k)=if(k==n, 1, 7*binomial(n, k)) \\ Charles R Greathouse IV, Jan 16 2012

(Magma) [k eq n select 1 else 7*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

(Sage)

@CachedFunction

def T(n, k):

if (k==n): return 1

else: return 7*binomial(n, k)

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

(GAP)

T:= function(n, k)

if k=n then return 1;

else return 7*Binomial(n, k);

fi; end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019

CROSSREFS

Cf. A007318, A048489, A131110, A131111, A131112, A131113, A131114.

KEYWORD

nonn,tabl,easy,less

AUTHOR

Gary W. Adamson, Jun 15 2007

EXTENSIONS

Corrected and extended by Emeric Deutsch, Jun 20 2007

STATUS

approved

A131112

T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

+10
6

1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1

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

OFFSET

0,2

LINKS

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

FORMULA

T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

n-th row sum = A036563(n+2) = 2^(n+2) - 3.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

4, 1;

4, 8, 1;

4, 12, 12, 1;

4, 16, 24, 16, 1;

4, 20, 40, 40, 20, 1;

...

MAPLE

seq(seq(`if`(k=n, 1, 4*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

MATHEMATICA

Table[If[k==n, 1, 4*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PROG

(PARI) T(n, k) = if(k==n, 1, 4*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019

(Magma) [k eq n select 1 else 4*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

(Sage)

def T(n, k):

if (k==n): return 1

else: return 4*binomial(n, k)

[[T(n, k) for k in (0..n)] for n in (0..10)]

# G. C. Greubel, Nov 18 2019

(GAP)

T:= function(n, k)

if k=n then return 1;

else return 4*Binomial(n, k);

fi; end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019

CROSSREFS

Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.

KEYWORD

nonn,tabl,easy,less

AUTHOR

Gary W. Adamson, Jun 15 2007

STATUS

approved

A131114

T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

+10
5

1, 6, 1, 6, 12, 1, 6, 18, 18, 1, 6, 24, 36, 24, 1, 6, 30, 60, 60, 30, 1, 6, 36, 90, 120, 90, 36, 1, 6, 42, 126, 210, 210, 126, 42, 1, 6, 48, 168, 336, 420, 336, 168, 48, 1, 6, 54, 216, 504, 756, 756, 504, 216, 54, 1

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

OFFSET

0,2

COMMENTS

Row sums give A048488.

LINKS

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

FORMULA

T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

6, 1;

6, 12, 1;

6, 18, 18, 1;

6, 24, 36, 24, 1;

6, 30, 60, 60, 30, 1;

6, 36, 90, 120, 90, 36, 1;

...

MAPLE

seq(seq(`if`(k=n, 1, 6*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

MATHEMATICA

Table[If[k==n, 1, 6*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PROG

(PARI) T(n, k) = if(k==n, 1, 6*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019

(Magma) [k eq n select 1 else 6*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

(Sage)

def T(n, k):

if (k==n): return 1

else: return 6*binomial(n, k)

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

(GAP)

T:= function(n, k)

if k=n then return 1;

else return 6*Binomial(n, k);

fi; end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019

CROSSREFS

Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.

KEYWORD

nonn,tabl,easy,less

AUTHOR

Gary W. Adamson, Jun 15 2007

STATUS

approved

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