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Search: a092398 -id:a092398
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A288580
Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.
+10
10
1, 1, -1, 1, -1, 4, 1, 1, -4, -36, 1, 1, -2, 9, 576, 1, 1, -4, -9, 64, -14400, 1, 1, 2, -3, -8, -225, 518400, 1, 1, 2, -6, -16, 40, -2304, -25401600, 1, 1, 2, -9, -4, -15, 324, 11025, 1625702400, 1, 1, 2, 3, -8, -25, 144, 280, 147456, -131681894400, 1, 1, 2, 3, -12, -5, -24, 105, -2240, -893025, 13168189440000
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=0..65.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
EXAMPLE
Array begins:
1, -1, 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400, ...
1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, ...
1, 1, -2, -9, -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, ...
1, 1, -4, -3, -16, -15, 144, 105, 1024, 945, -14400, -10395, -147456, ...
1, 1, 2, -6, -4, -25, -24, -42, 336, 216, 2500, 2376, 4032, ...
1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, ...
1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, ...
1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, ...
1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, ...
1, 1, 2, 3, 4, -25, -24, -21, -16, -9, -100, -99, -192, ...
...
MAPLE
T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
scan1:=proc(a, M1) local lis, n, k; lis:=[]; for n from 1 to M1 do for k from 0 to n-1 do
lis:=[op(lis), a(k, n-k)]; od: od: lis; end:
scan1(T, 12);
MATHEMATICA
T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;
Table[T[k, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 05 2020, after Maple *)
CROSSREFS
Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092971
Row 6 of array in A288580.
+10
7
1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, 5005, 8960, -164025, -143360, -85085, -1679616, -1616615, -2867200, 72335025, 63078400, 37182145, 967458816, 929553625, 1640038400, -52732233225, -45921075200, -26957055125, -870712934400, -835668708875, -1469474406400, 57425401982025
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=0..33.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
FORMULA
a(n, k) = !n!_k = Prod_{i=0, 1, 2, .., floor(2n/k)}_{0<|n-i*k|<=n} (n-i*k) = n(n-k)(n-2k)(n-3k)... . k=6.
MAPLE
T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n, k), n=0..60)]; r(6); # N. J. A. Sloane, Jul 03 2017
PROG
(PARI) a(n, k)=prod(j=0, (2*n)\k, if(n-k*j==0, 1, n-k*j))
CROSSREFS
Cf. A288580, A092396, A092397, A092398, A092399, A092972, A092973, A092974.
KEYWORD
sign
AUTHOR
Paul D. Hanna and Amarnath Murthy, Mar 27 2004
EXTENSIONS
Entry revised by N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092972
Row 7 of array in A288580.
+10
7
1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, 624, 9604, 9360, 17280, 22440, -403920, -328320, -187200, -4235364, -4118400, -7551360, -9694080, 242352000, 196335360, 111196800, 3320525376, 3224707200, 5890060800, 7512912000, -240413184000, -194372006400, -109640044800
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=0..34.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
FORMULA
a(n, k) = !n!_k = Prod_{i=0, 1, 2, .., floor(2n/k)}_{0<|n-i*k|<=n} (n-i*k) = n(n-k)(n-2k)(n-3k)... . k=7.
MAPLE
T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n, k), n=0..60)]; r(7); # N. J. A. Sloane, Jul 03 2017
PROG
(PARI) a(n, k)=prod(j=0, (2*n)\k, if(n-k*j==0, 1, n-k*j))
CROSSREFS
Cf. A288580, A092396, A092397, A092398, A092399, A092971, A092973, A092974.
KEYWORD
sign
AUTHOR
Paul D. Hanna, M.L. Perez and Amarnath Murthy, Mar 27 2004
EXTENSIONS
Entry revised by N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092973
Row 8 of array in A288580.
+10
7
1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, 2145, 1680, 945, 16384, 16065, 30240, 40755, -921600, -855855, -665280, -369495, -9437184, -9237375, -17297280, -23108085, 722534400, 670134465, 518918400, 286358625, 9663676416, 9449834625, 17643225600, 23454706275, -936404582400
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=0..36.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
FORMULA
a(n, k) = !n!_k = Prod_{i=0, 1, 2, .., floor(2n/k)}_{0<|n-i*k|<=n} (n-i*k) = n(n-k)(n-2k)(n-3k)... . k=8.
MAPLE
T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n, k), n=0..60)]; r(8); # N. J. A. Sloane, Jul 03 2017
PROG
(PARI) a(n, k)=prod(j=0, (2*n)\k, if(n-k*j==0, 1, n-k*j))
CROSSREFS
Cf. A288580, A092396, A092397, A092398, A092399, A092971, A092972, A092974.
KEYWORD
sign
AUTHOR
Paul D. Hanna, M.L. Perez and Amarnath Murthy, Mar 27 2004
EXTENSIONS
Entry revised by N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092974
Row 9 of array in A288580.
+10
7
1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, -260, 3640, 3240, 2464, 1360, 26244, 25840, 49280, 68040, 80080, -1841840, -1632960, -1232000, -671840, -19131876, -18811520, -35728000, -48988800, -57097040, 1827105280, 1616630400, 1214752000, 658403200, 24794911296, 24360918400
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=0..37.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
FORMULA
a(n, k) = !n!_k = Prod_{i=0, 1, 2, .., floor(2n/k)}_{0<|n-i*k|<=n} (n-i*k) = n(n-k)(n-2k)(n-3k)... . k=9.
MAPLE
T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n, k), n=0..60)]; r(9); # N. J. A. Sloane, Jul 03 2017
PROG
(PARI) a(n, k)=prod(j=0, (2*n)\k, if(n-k*j==0, 1, n-k*j))
CROSSREFS
Cf. A288580, A092396, A092397, A092398, A092399, A092971, A092972, A092973.
KEYWORD
sign
AUTHOR
Paul D. Hanna, M.L. Perez and Amarnath Murthy, Mar 27 2004
STATUS
approved
A092096
a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=5.
+10
2
11, 12, 20, 20, 30, 31, 32, 45, 45, 60, 61, 62, 80, 80, 100, 101, 102, 125, 125, 150, 151, 152, 180, 180, 210, 211, 212, 245, 245, 280, 281, 282, 320, 320, 360, 361, 362, 405, 405, 450, 451, 452, 500, 500, 550, 551, 552, 605, 605, 660, 661, 662, 720, 720, 780
(list; graph; refs; listen; history; text; internal format)
OFFSET
6,1
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=6..60.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
F. Smarandache, Summants [Broken link]
FORMULA
Empirical g.f.: -x^6*(10*x^10-5*x^9-3*x^7-x^6-21*x^5+10*x^4+8*x^2+x+11) / ((x-1)^3*(x^4+x^3+x^2+x+1)^2). - Colin Barker, Jul 28 2013
MAPLE
S := proc(n, k) local a, i ; a :=0 ; i := 0 ; while n-k*i >= -n do a := a+abs(n-k*i) ; i := i+1 ; od: RETURN(a) ; end: k := 5: seq(S(n, k), n=k+1..80) ; # R. J. Mathar, Feb 01 2008
MATHEMATICA
a[n_] := Sum[Abs[n-5i], {i, 0, Quotient[2n, 5]}];
Table[a[n], {n, 6, 60}] (* Jean-François Alcover, Apr 29 2023 *)
CROSSREFS
Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.
KEYWORD
nonn
AUTHOR
Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004
EXTENSIONS
Edited and extended by R. J. Mathar, Feb 01 2008
Revised by N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092094
a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=3.
+10
1
7, 12, 18, 19, 27, 36, 37, 48, 60, 61, 75, 90, 91, 108, 126, 127, 147, 168, 169, 192, 216, 217, 243, 270, 271, 300, 330, 331, 363, 396, 397, 432, 468, 469, 507, 546, 547, 588, 630, 631, 675, 720, 721, 768, 816, 817, 867, 918, 919, 972, 1026, 1027, 1083, 1140
(list; graph; refs; listen; history; text; internal format)
OFFSET
4,1
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=4..57.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
F. Smarandache, Summants [Broken link]
FORMULA
S_abs(n, 3) = Sigma_{i=0, 1, 2, ...}_{0<abs(n-3i)<=n}(abs(n-3i)) = n+abs(n-3)+abs(n-6)+ ...
Empirical g.f.: -x^4*(6*x^6-3*x^5-2*x^4-13*x^3+6*x^2+5*x+7) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Jul 28 2013
EXAMPLE
S_abs(7, 3) = 7+abs(7-3)+abs(7-6)+abs(7-9)+abs(7-12) = 7+4+1+2+5 = 19.
MAPLE
S := proc(n, k) local a, i ; a :=0 ; i := 0 ; while n-k*i >= -n do a := a+abs(n-k*i) ; i := i+1 ; od: RETURN(a) ; end: k := 3: seq(S(n, 3), n=k+1..80) ; # R. J. Mathar, Feb 01 2008
MATHEMATICA
S[n_, k_] := Module[{a = 0, i = 0}, While[n - k i >= -n, a += Abs[n - k i]; i++]; a];
Table[S[n, 3], {n, 4, 80}] (* Jean-François Alcover, Apr 05 2020, from Maple *)
CROSSREFS
Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.
KEYWORD
nonn
AUTHOR
Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004
EXTENSIONS
Edited and extended by R. J. Mathar, Feb 01 2008
Definition clarified by N. J. A. Sloane, Jul 03 2017
STATUS
approved
A092095
a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=4.
+10
0
9, 16, 16, 24, 25, 36, 36, 48, 49, 64, 64, 80, 81, 100, 100
(list; graph; refs; listen; history; text; internal format)
OFFSET
4,1
REFERENCES
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.
LINKS
Table of n, a(n) for n=4..18.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
F. Smarandache, Summants [Broken link]
MAPLE
S := proc(n, k) local a, i ; a :=0 ; i := 0 ; while n-k*i >= -n do a := a+abs(n-k*i) ; i := i+1 ; od: RETURN(a) ; end: k := 4: seq(S(n, k), n=k+1..80) ; # R. J. Mathar, Feb 01 2008 (Adapted from program for A092096 by N. J. A. Sloane, Jul 03 2017)
CROSSREFS
Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.
KEYWORD
nonn,more
AUTHOR
Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004
EXTENSIONS
Edited with better definition by Omar E. Pol, Dec 28 2008
Entry revised by N. J. A. Sloane, Jul 03 2017
STATUS
approved
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