A174377 -id:A174377

Search: a174377 -id:a174377

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A159623

Triangle read by rows: T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 1.

+10
4

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 12, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 30, 120, 30, 6, 1, 1, 7, 42, 210, 210, 42, 7, 1, 1, 8, 56, 336, 1680, 336, 56, 8, 1, 1, 9, 72, 504, 3024, 3024, 504, 72, 9, 1, 1, 10, 90, 720, 5040, 30240, 5040, 720, 90, 10, 1

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

OFFSET

0,5

COMMENTS

The row sums are: {1, 2, 4, 8, 22, 52, 194, 520, 2482, 7220, 41962,...}.

LINKS

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

FORMULA

T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 1.

T(n, n-k) = T(n, k).

T(2*n, n) = A001813(n). - G. C. Greubel, Nov 28 2021

EXAMPLE

Triangle begins as:

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 12, 4, 1;

1, 5, 20, 20, 5, 1;

1, 6, 30, 120, 30, 6, 1;

1, 7, 42, 210, 210, 42, 7, 1;

1, 8, 56, 336, 1680, 336, 56, 8, 1;

1, 9, 72, 504, 3024, 3024, 504, 72, 9, 1;

1, 10, 90, 720, 5040, 30240, 5040, 720, 90, 10, 1;

MATHEMATICA

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];

Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Sage)

f=factorial

def T(n, k, q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)

flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

CROSSREFS

Cf. this sequence (q=1), A174376 (q=2), A174377 (q=3), A174378 (q=4).

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Apr 17 2009

EXTENSIONS

Edited by N. J. A. Sloane, Apr 17 2009

STATUS

approved

A174376

Triangle T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2, read by rows.

+10
4

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 48, 8, 1, 1, 10, 80, 80, 10, 1, 1, 12, 120, 960, 120, 12, 1, 1, 14, 168, 1680, 1680, 168, 14, 1, 1, 16, 224, 2688, 26880, 2688, 224, 16, 1, 1, 18, 288, 4032, 48384, 48384, 4032, 288, 18, 1, 1, 20, 360, 5760, 80640, 967680, 80640, 5760, 360, 20, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 6, 14, 66, 182, 1226, 3726, 32738, 105446, 1141242, ...}.

LINKS

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

FORMULA

T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2.

T(n, n-k) = T(n, k).

T(2*n, n) = A052714(n+1). - G. C. Greubel, Nov 28 2021

EXAMPLE

Triangle begins as:

1, 1;

1, 4, 1;

1, 6, 6, 1;

1, 8, 48, 8, 1;

1, 10, 80, 80, 10, 1;

1, 12, 120, 960, 120, 12, 1;

1, 14, 168, 1680, 1680, 168, 14, 1;

1, 16, 224, 2688, 26880, 2688, 224, 16, 1;

1, 18, 288, 4032, 48384, 48384, 4032, 288, 18, 1;

1, 20, 360, 5760, 80640, 967680, 80640, 5760, 360, 20, 1;

MATHEMATICA

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Sage)

f=factorial

def T(n, k, q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)

flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

CROSSREFS

Cf. A159623 (q=1), this sequence (q=2), A174377 (q=3), A174378 (q=4).

Cf. A052714.

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Mar 17 2010

EXTENSIONS

Edited by G. C. Greubel, Nov 28 2021

STATUS

approved

A174378

Triangle T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 4, read by rows.

+10
4

1, 1, 1, 1, 8, 1, 1, 12, 12, 1, 1, 16, 192, 16, 1, 1, 20, 320, 320, 20, 1, 1, 24, 480, 7680, 480, 24, 1, 1, 28, 672, 13440, 13440, 672, 28, 1, 1, 32, 896, 21504, 430080, 21504, 896, 32, 1, 1, 36, 1152, 32256, 774144, 774144, 32256, 1152, 36, 1, 1, 40, 1440, 46080, 1290240, 30965760, 1290240, 46080, 1440, 40, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 10, 26, 226, 682, 8690, 28282, 474946, 1615178, ...}.

LINKS

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

FORMULA

T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2.

T(n, n-k) = T(n, k).

T(2*n, n) = A052734(n+1). - G. C. Greubel, Nov 28 2021

EXAMPLE

Triangle begins as:

1, 1;

1, 8, 1;

1, 12, 12, 1;

1, 16, 192, 16, 1;

1, 20, 320, 320, 20, 1;

1, 24, 480, 7680, 480, 24, 1;

1, 28, 672, 13440, 13440, 672, 28, 1;

1, 32, 896, 21504, 430080, 21504, 896, 32, 1;

1, 36, 1152, 32256, 774144, 774144, 32256, 1152, 36, 1;

1, 40, 1440, 46080, 1290240, 30965760, 1290240, 46080, 1440, 40, 1;

MATHEMATICA

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];

Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Sage)

f=factorial

def T(n, k, q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)

flatten([[T(n, k, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

CROSSREFS

Cf. A159623 (q=1), A174376 (q=2), A174377 (q=3), this sequence (q=4).

Cf. A052734.

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Mar 17 2010

EXTENSIONS

Edited by G. C. Greubel, Nov 28 2021

STATUS

approved

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