cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Jul 03 2017

Keywords

Examples

			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, ...
...

References

  • F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

Links

Crossrefs

Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,

Programs

  • 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 *)

A092971 Row 6 of array in A288580.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna and Amarnath Murthy, Mar 27 2004

Keywords

References

  • F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

Links

Crossrefs

Cf. A288580, A092396, A092397, A092398, A092399, A092972, A092973, A092974.

Programs

  • 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
  • PARI
    a(n,k)=prod(j=0,(2*n)\k,if(n-k*j==0,1,n-k*j))

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.

Extensions

Entry revised by N. J. A. Sloane, Jul 03 2017

A092972 Row 7 of array in A288580.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, M.L. Perez and Amarnath Murthy, Mar 27 2004

Keywords

References

  • F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

Links

Crossrefs

Cf. A288580, A092396, A092397, A092398, A092399, A092971, A092973, A092974.

Programs

  • 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
  • PARI
    a(n,k)=prod(j=0,(2*n)\k,if(n-k*j==0,1,n-k*j))

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.

Extensions

Entry revised by N. J. A. Sloane, Jul 03 2017

A092974 Row 9 of array in A288580.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, M.L. Perez and Amarnath Murthy, Mar 27 2004

Keywords

References

  • F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

Links

Crossrefs

Cf. A288580, A092396, A092397, A092398, A092399, A092971, A092972, A092973.

Programs

  • 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
  • PARI
    a(n,k)=prod(j=0,(2*n)\k,if(n-k*j==0,1,n-k*j))

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.

A092096 a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=5.

Original entry on oeis.org

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
Offset: 6

Views

Author

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

Keywords

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

Crossrefs

Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.

Programs

  • 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 *)

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

Extensions

Edited and extended by R. J. Mathar, Feb 01 2008
Revised by N. J. A. Sloane, Jul 03 2017

A092094 a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=3.

Original entry on oeis.org

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
Offset: 4

Views

Author

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

Keywords

Examples

			S_abs(7, 3) = 7+abs(7-3)+abs(7-6)+abs(7-9)+abs(7-12) = 7+4+1+2+5 = 19.

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

Crossrefs

Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.

Programs

  • 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 *)

Formula

S_abs(n, 3) = Sigma_{i=0, 1, 2, ...}_{0
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

Extensions

Edited and extended by R. J. Mathar, Feb 01 2008
Definition clarified by N. J. A. Sloane, Jul 03 2017

A092095 a(n) = Sum_{i=0,1,2,...; n-k*i >= -n} |n-k*i| for k=4.

Original entry on oeis.org

9, 16, 16, 24, 25, 36, 36, 48, 49, 64, 64, 80, 81, 100, 100, 120, 121, 144, 144, 168, 169, 196, 196, 224, 225, 256, 256, 288, 289, 324, 324, 360, 361, 400, 400, 440, 441, 484, 484, 528, 529, 576, 576, 624, 625, 676, 676, 728, 729, 784, 784, 840, 841, 900, 900
Offset: 5

Views

Author

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

Keywords

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

Crossrefs

Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.

Programs

  • 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)
  • PARI
    a(n) = ((2*n+1)*(-1)^n - 2*(-I)^n - 2*I^n + 2*n*(n+3) + 3)/8; \\ Jinyuan Wang, Apr 09 2025

Formula

G.f.: x^5*(8*x^6-4*x^5-8*x^4+x^3-9*x^2+7*x+9)/((x^2+1)*(x+1)^2*(1-x)^3). - Alois P. Heinz, Apr 09 2025

Extensions

Edited with better definition by Omar E. Pol, Dec 28 2008
Entry revised by N. J. A. Sloane, Jul 03 2017
Offset changed to 5 and more terms from Jinyuan Wang, Apr 09 2025
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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