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.

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A174295 Matrix inverse of A174294.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 1, -2, -1, 1, 0, 1, -3, 2, 0, -2, 2, 0, 1, 6, -7, -3, 3, -3, 3, 0, 1, -15, 14, 3, -10, 7, -4, 4, 0, 1, 36, -37, -12, 19, -19, 12, -5, 5, 0, 1, -91, 90, 24, -54, 42, -30, 18, -6, 6, 0, 1, 232, -233, -67, 127, -115, 73, -43, 25, -7, 7
Offset: 0

Views

Author

Mats Granvik, Mar 15 2010

Keywords

Comments

First column is a signed version of A099323 with an additional leading 1.
First 5 rows as in A054525.

Examples

			Table begins:
  n\k|...0...1...2...3...4...5...6...7...8...9..10
  ---|--------------------------------------------
  0..|...1
  1..|..-1...1
  2..|..-1...0...1
  3..|...0..-1...0...1
  4..|..-1...0...0...0...1
  5..|...1..-2..-1...1...0...1
  6..|..-3...2...0..-2...2...0...1
  7..|...6..-7..-3...3..-3...3...0...1
  8..|.-15..14...3.-10...7..-4...4...0...1
  9..|..36.-37.-12..19.-19..12..-5...5...0...1
  10.|.-91..90..24.-54..42.-30..18..-6...6...0...1

Links

Crossrefs

Cf. A000007 (row sums), A054525, A099323, A112467, A112468, A174294, A174296, A174297.

Programs

  • Mathematica
    t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], t[n-1, k-1] +t[n-2, k-1] -t[n-1, k] -t[n-2, k] ]]]; (* t = A174294 *)
M:= With[{m=30}, Table[t[n, k], {n,0,m}, {k,0,m}]];
T:= Inverse[M];
Table[T[[n+1, k+1]], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 25 2021 *)

Formula

Sum_{k=0..n} T(n, k) = A000007(n).
T(n, 0) = A174297(n).

A174297 First column of A174295.

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -3, 6, -15, 36, -91, 232, -603, 1585, -4213, 11298, -30537, 83097, -227475, 625992, -1730787, 4805595, -13393689, 37458330, -105089229, 295673994, -834086421, 2358641376, -6684761125, 18985057351, -54022715451
Offset: 0

Views

Author

Mats Granvik, Mar 15 2010

Keywords

Comments

First 6 terms as in Mobius function A008683. Signed version of A099323 with an additional leading 1.

Links

Crossrefs

Cf. 099323, A112467, A112468, A174294, A174295, A174296.

Programs

  • Magma
    a:= func< n | n lt 2 select (-1)^n else (&+[(-1)^(k+1)*Binomial(n-2, k)*Catalan(k): k in [0..n-2]]) >;
[a(n): n in [0..30]]; // G. C. Greubel, Nov 25 2021
  • Mathematica
    a[n_]:= a[n]= If[n<2, (-1)^n, Sum[(-1)^(j+1)*Binomial[n-2, j]*CatalanNumber[j], {j, 0, n-2}]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Nov 25 2021 *)
  • Sage
    [1,-1]+[sum( (-1)^(j+1)*binomial(n-2,j)*catalan_number(j) for j in (0..n-2) ) for n in (2..40)] # G. C. Greubel, Nov 25 2021

Formula

a(n) = -(-3)^(n-3/2)*hypergeometric2F1([3/2, n-1],[2],4) for n > 2. - Mark van Hoeij, Jul 02 2010
a(n) = (-1)^n if n < 2 otherwise Sum_{j=0..n-2} (-1)^(j-1)*binomial(n-2, j)*Catalan(j). - G. C. Greubel, Nov 25 2021

A238275 a(n) = (4*7^n - 1)/3.

Original entry on oeis.org

1, 9, 65, 457, 3201, 22409, 156865, 1098057, 7686401, 53804809, 376633665, 2636435657, 18455049601, 129185347209, 904297430465, 6330082013257, 44310574092801, 310174018649609, 2171218130547265, 15198526913830857, 106389688396816001, 744727818777712009
Offset: 0

Views

Author

Philippe Deléham, Feb 21 2014

Keywords

Comments

Sum of n-th row of triangle of powers of 7: 1; 1 7 1; 1 7 49 7 1; 1 7 49 343 49 7 1; ...
Number of cubes in the crystal structure cubic carbon CCC(n+1), defined in the Baig et al. and in the Gao et al. references. - Emeric Deutsch, May 28 2018

Examples

			a(0) = 1;
a(1) = 1 + 7 + 1 = 9;
a(2) = 1 + 7 + 49 + 7 + 1 = 65;
a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.

Links

Crossrefs

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, this sequence, A238276, A138894, A090843, A199023.
Cf. A112468, A112739.

Programs

Formula

G.f.: (1+x)/((1-x)*(1-7*x)).
a(n) = 7*a(n-1) + 2, a(0) = 1.
a(n) = 8*a(n-1) - 7*a(n-2), a(0) = 1, a(1) = 9.
a(n) = Sum_{k=0..n} A112468(n,k)*8^k.
E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - Stefano Spezia, Feb 12 2025

A238276 a(n) = (9*8^n - 2)/7.

Original entry on oeis.org

1, 10, 82, 658, 5266, 42130, 337042, 2696338, 21570706, 172565650, 1380525202, 11044201618, 88353612946, 706828903570, 5654631228562, 45237049828498, 361896398627986, 2895171189023890, 23161369512191122, 185290956097528978, 1482327648780231826
Offset: 0

Views

Author

Philippe Deléham, Feb 21 2014

Keywords

Comments

Sum of n-th row of triangle of powers of 8: 1; 1 8 1; 1 8 64 8 1; 1 8 64 512 64 8 1; ...

Examples

			a(0) = 1;
a(1) = 1 + 8 + 1 = 10;
a(2) = 1 + 8 + 64 + 8 + 1 = 82;
a(3) = 1 + 8 + 64 + 512 + 64 + 8 + 1 = 658; etc.

Links

Crossrefs

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, this sequence, A138894, A090843, A199023.
Cf. A112468, A112739.

Programs

Formula

G.f.: (1+x)/((1-x)*(1-8*x)).
a(n) = 8*a(n-1) + 2, a(0) = 1.
a(n) = 9*a(n-1) - 8*a(n-2), a(0) = 1, a(1) = 10.
a(n) = Sum_{k=0..n} A112468(n,k)*9^k.

Extensions

Corrected by Vincenzo Librandi, Feb 23 2014

A112469 Partial sums of (-1)^n*Fibonacci(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, -3, 10, -11, 23, -32, 57, -87, 146, -231, 379, -608, 989, -1595, 2586, -4179, 6767, -10944, 17713, -28655, 46370, -75023, 121395, -196416, 317813, -514227, 832042, -1346267, 2178311, -3524576, 5702889, -9227463, 14930354, -24157815, 39088171, -63245984, 102334157
Offset: 0

Views

Author

Paul Barry, Sep 06 2005

Keywords

Comments

Diagonal sums of Riordan array (1/(1-x), x/(1+x)), A112468.

Links

Crossrefs

Cf. A000045, A078024, A112468.

Programs

  • Magma
    A112469:= func< n | 2 + (-1)^n*Fibonacci(n-2) >;
[A112469(n): n in [0..40]]; // G. C. Greubel, Apr 17 2025
  • Maple
    a[0]:=1:a[1]:=1:a[2]:=2:a[3]:=1:for n from 4 to 50 do a[n]:=2*a[n-2]-a[n-3] od: seq(a[n], n=0..42); # Zerinvary Lajos, Apr 04 2008
  • Mathematica
    Accumulate[Table[(-1)^n Fibonacci[n-1],{n,0,50}]] (* Harvey P. Dale, Nov 05 2011 *)
Table[2 +(-1)^n*Fibonacci[n-2], {n,0,50}] (* G. C. Greubel, Apr 17 2025 *)
  • SageMath
    def A112469(n): return 2+(-1)^n*fibonacci(n-2)
print([A112469(n) for n in range(41)]) # G. C. Greubel, Apr 17 2025

Formula

G.f.: (1+x)/((1-x)*(1+x-x^2)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} C(n-k-j-1, n-2k-j)*(-1)^(n-j).
From G. C. Greubel, Apr 17 2025: (Start)
a(n) = 2 + (-1)^n*Fibonacci(n-2).
E.g.f.: 2*exp(x) - exp(-x/2)*( cosh(sqrt(5)*x/2) + (3/sqrt(5))*sinh(sqrt(5)*x/2) ). (End)

A279006 Alternating Jacobsthal triangle read by rows (second version).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, -2, 2, 0, 1, 1, -3, 4, -2, 1, 1, 1, -4, 7, -6, 3, 0, 1, 1, -5, 11, -13, 9, -3, 1, 1, 1, -6, 16, -24, 22, -12, 4, 0, 1, 1, -7, 22, -40, 46, -34, 16, -4, 1, 1, 1, -8, 29, -62, 86, -80, 50, -20, 5, 0, 1, 1, -9, 37, -91, 148, -166, 130, -70, 25, -5, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Examples

			Triangle begins:
  1,
  1,   1,
  1,   0,   1,
  1,  -1,   1,   1,
  1,  -2,   2,   0,   1,
  1,  -3,   4,  -2,   1,   1,
  1,  -4,   7,  -6,   3,   0,   1,
  1,  -5,  11, -13,   9,  -3,   1,   1,
  1,  -6,  16, -24,  22, -12,   4,   0,   1,
  ...

Links

Crossrefs

See A112468, A112555 and A108561 for other versions.
Columns give A000124, A003600, A223718, A257890, A223659.

Programs

  • Maple
    T := (n, k) -> local j; add((-1)^j*binomial(n-k-1+j, j), j = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, Aug 30 2024
  • Mathematica
    T[i_, i_] = T[, 0] = 1; T[i, j_] := T[i, j] = T[i-1, j] - T[i-1, j-1];
Table[T[i, j], {i, 0, 11}, {j, 0, i}] // Flatten (* Jean-François Alcover, Sep 06 2018 *)
T[n_, k_] := 2^k*Hypergeometric2F1[-n, -k, -k, 1/2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Detlef Meya, Aug 30 2024 *)
  • PARI
    \\ using arxiv (3.1) and (3.7) formulas where A is A220074 and B is this sequence
A(i, j) = if ((i < 0), 0, if (j==0, 1, A(i - 1, j - 1) - A(i - 1, j))); \\ A220074
B(i, j) = A(i, i-j);
tabl(nn) = for (i=0, nn, for (j=0, i, print1(B(i,j), ", ")); print()); \\ Michel Marcus, Jun 17 2017

Formula

T(i, j) = A220074(i, i-j). See (3.7) in arxiv link. - Michel Marcus, Jun 17 2017
T(n, k) = 2^k*hypergeom([-n, -k], [-k], 1/2). - Detlef Meya, Aug 30 2024

Extensions

More terms from Michel Marcus, Jun 17 2017

A279009 Alternating Jacobsthal triangle A_{-2}(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, -2, 0, 1, -2, -2, -1, 1, 4, 0, -1, -2, 1, 4, 4, 1, 1, -3, 1, -8, 0, 3, 0, 4, -4, 1, -8, -8, -3, 3, -4, 8, -5, 1, 16, 0, -5, -6, 7, -12, 13, -6, 1, 16, 16, 5, 1, -13, 19, -25, 19, -7, 1, -32, 0, 11, 4, 14, -32, 44, -44, 26, -8, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Examples

			Triangle begins:
1,
1,   1,
-2,  0,  1,
-2, -2, -1,  1,
4,   0, -1, -2,   1,
4,   4,  1,  1,  -3,   1,
-8,  0,  3,  0,   4, - 4,   1,
-8, -8, -3,  3,  -4,   8,  -5,   1,
16,  0, -5, -6,   7, -12,  13,  -6,  1,
16, 16,  5,  1, -13,  19, -25,  19, -7,  1,
-32, 0, 11,  4,  14, -32,  44, -44, 26, -8, 1,
...

Links

Crossrefs

Cf. A112468, A279006, A279010.

Programs

  • Mathematica
    a[n_, 0] := (-2)^Floor[n/2]; a[n_, n_] = 1; a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n-1, k-1] - a[n-1, k]; a[, ] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 05 2018 *)

A173433 a(n) = (A000045(n)+A173432(n))/2.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 7, 11, 18, 28, 45, 72, 117, 189, 306, 494, 799, 1292, 2091, 3383, 5474, 8856, 14329, 23184, 37513, 60697, 98210, 158906, 257115, 416020, 673135, 1089155, 1762290, 2851444, 4613733, 7465176, 12078909, 19544085, 31622994, 51167078, 82790071, 133957148
Offset: 1

Views

Author

Mark Dols, Feb 18 2010

Keywords

Comments

Also the NW-SE diagonal sums of A173398.

Links

Crossrefs

Cf. A112468, A000045, A057079, A173432, A173398.

Programs

  • Maple
    f:=gfun:-rectoproc({-a(n) - a(n + 1) + a(n + 2) - a(n + 3) - a(n + 4) + a(n + 5) + 1, a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 2, a(5) = 3},a(n),remember):
map(f, [$1..50]); # Robert Israel, Jun 11 2019
  • Mathematica
    CoefficientList[Series[-x*(-1+x+x^4)/((x-1)*(1+x)*(x^2+x-1)*(x^2-x+1)),{x,0,42}],x] (* Georg Fischer, Jun 11 2019 *)

Formula

a(n) = 1/2-(-1)^n/6+A057079(n+4)/6+A000045(n)/2 with g.f. -x*(-1+x+x^4)/ ((x-1) * (1+x) * (x^2+x-1) * (x^2-x+1)). - R. J. Mathar, Mar 04 2010

Extensions

a(35) corrected by Georg Fischer, Jun 11 2019

A279010 Alternating Jacobsthal triangle A_3(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, -1, 1, 9, 0, 4, -2, 1, 9, 9, -4, 6, -3, 1, 27, 0, 13, -10, 9, -4, 1, 27, 27, -13, 23, -19, 13, -5, 1, 81, 0, 40, -36, 42, -32, 18, -6, 1, 81, 81, -40, 76, -78, 74, -50, 24, -7, 1, 243, 0, 121, -116, 154, -152, 124, -74, 31, -8, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Examples

			Triangle begins:
    1;
    1,  1;
    3,  0,   1;
    3,  3,  -1,    1;
    9,  0,   4,   -2,   1;
    9,  9,  -4,    6,  -3,    1;
   27,  0,  13,  -10,   9,   -4,   1;
   27, 27, -13,   23, -19,   13,  -5,   1;
   81,  0,  40,  -36,  42,  -32,  18,  -6,  1;
   81, 81, -40,   76, -78,   74, -50,  24, -7,  1;
  243,  0, 121, -116, 154, -152, 124, -74, 31, -8, 1;
  ...

Links

Crossrefs

If initial column is omitted, this is very like the Riordan matrix A191582.
Cf. A112468, A279006, A279009.

Programs

  • Mathematica
    A[n_, 0] := 3^Floor[n/2];
A[n_, k_] /; (k<0 || t>n) = 0;
A[n_, n_] = 1;
A[n_, k_] := A[n, k] = A[n-1, k-1] - A[n-1, k];
Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

A173434 a(n) = (A000045(n)-A173432(n))/2.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 10, 16, 27, 44, 72, 116, 188, 304, 493, 798, 1292, 2090, 3382, 5472, 8855, 14328, 23184, 37512, 60696, 98208, 158905, 257114, 416020, 673134, 1089154, 1762288, 2851443, 4613732, 7465176
Offset: 1

Views

Author

Mark Dols, Feb 18 2010

Keywords

Comments

Also the NW-SE diagonal sums of A173402.

Links

Crossrefs

Cf. A112468, A000045, A173398, A173402

Programs

  • Mathematica
    CoefficientList[Series[x^4/((x-1)(1+x)(x^2-x+1)(x^2+x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,-2,2,0,-1},{0,0,0,0,1,2},40] (* Harvey P. Dale, Jun 29 2021 *)

Formula

a(n) + A173433(n) = A000045(n).
a(n)= 2*a(n-1) -2*a(n-3) +2*a(n-4) -a(n-6). - R. J. Mathar, Mar 01 2010
G.f.: x^4 / ( (x-1)*(1+x)*(x^2-x+1)*(x^2+x-1) ). - R. J. Mathar, Nov 03 2016
Previous Showing 21-30 of 30 results.

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

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