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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A080957 Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.

Original entry on oeis.org

5, 11, 20, 34, 55, 85, 126, 180, 249, 335, 440, 566, 715, 889, 1090, 1320, 1581, 1875, 2204, 2570, 2975, 3421, 3910, 4444, 5025, 5655, 6336, 7070, 7859, 8705, 9610, 10576, 11605, 12699, 13860, 15090, 16391, 17765, 19214, 20740, 22345, 24031, 25800
Offset: 0

Views

Author

Paul Barry, Mar 01 2003

Keywords

Comments

Coefficient of x in the polynomial 6*(C(n,0) + C(n+1,1)*x + C(n+2,2)*x*(x-1)/2 + C(n+3,3)*x*(x-1)*(x-2)/6).

Links

Crossrefs

Cf. A080956, A117951.

Programs

  • Magma
    [(2*n^3+3*n^2+31*n+30)/6: n in [0..50]]; // Vincenzo Librandi, Sep 07 2015
  • Mathematica
    CoefficientList[Series[(5-9 x +6 x^2)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi Sep 07 2015 *)
LinearRecurrence[{4,-6,4,-1},{5,11,20,34},50] (* Harvey P. Dale, Dec 23 2018 *)
  • PARI
    Vec((5-9*x+6*x^2)/(1-x)^4 + O(x^60)) \\ Michel Marcus, Sep 06 2015
  • PARI
    a(n)=(2*n^3 + 3*n^2 + 31*n + 30)/6;
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 28 2015
  • SageMath
    def A080957(n): return (2*n^3 +3*n^2 +31*n +30)//6
print([A080957(n) for n in range(51)]) # G. C. Greubel, May 08 2025

Formula

a(n) = 3!*(C(n+1, 1) - C(n+2, 2)/2 + C(n+3, 3)/3) = (2*n^3 + 3*n^2 + 31*n + 30)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Vincenzo Librandi, Sep 07 2015
a(n+1) = a(n) + A117951(n+1), a(0) = 5. - Altug Alkan, Sep 28 2015
E.g.f.: (1/6)*(30 + 36*x + 9*x^2 + 2*x^3)*exp(x). - G. C. Greubel, May 08 2025

A081498 Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,... up to n-1. Sequence gives row sums.

Original entry on oeis.org

1, 3, 5, 6, 5, 1, -7, -20, -39, -65, -99, -142, -195, -259, -335, -424, -527, -645, -779, -930, -1099, -1287, -1495, -1724, -1975, -2249, -2547, -2870, -3219, -3595, -3999, -4432, -4895, -5389, -5915, -6474, -7067, -7695, -8359, -9060, -9799, -10577, -11395, -12254, -13155, -14099, -15087, -16120
Offset: 1

Views

Author

Amarnath Murthy, Mar 25 2003

Keywords

Comments

The triangle whose row sums are being considered is:
1;
2, 1;
3, 2, 0;
4, 3, 1, -2;
5, 4, 2, -1, -5;
6, 5, 3, 0, -4, -9;
7, 6, 4, 1, -3, -8, -14;
The leading diagonal is given by A080956(n-1) = n*(3-n)/2.

Examples

			G.f. = x * (1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + x^5 - 7*x^6 - 20*x^7 - 39*x^8 - 65*x^9 + ...).

Links

Crossrefs

Cf. A008778, A080956, A081499.

Programs

  • GAP
    List([1..50],n->n^2-Binomial(n+1,n-2)); # Muniru A Asiru, Mar 05 2019
  • Magma
    [n*(1+6*n-n^2)/6: n in [1..50]]; // G. C. Greubel, Mar 06 2019
  • Maple
    seq(n^2-binomial(n+1,n-2),n=1..50); # C. Ronaldo
[seq(binomial(n,2)+binomial(n,1)-binomial(n,3), n=1..49)]; # Zerinvary Lajos, Jul 23 2006
  • Mathematica
    LinearRecurrence[{4,-6,4,-1}, {1,3,5,6}, 50] (* G. C. Greubel, Mar 06 2019 *)
  • PARI
    {a(n) = if( n< 0, n = -2 - n; polcoeff( (1 + x - x^2) / (1 - x)^4 + x * O(x^n), n), polcoeff( (1 - x - x^2) / (1 - x)^4 + x * O(x^n), n))} /* Michael Somos, Jul 04 2012 */
  • PARI
    vector(50, n, n*(1+6*n-n^2)/6) \\ G. C. Greubel, Mar 06 2019
  • Sage
    [n*(1+6*n-n^2)/6 for n in (1..50)] # G. C. Greubel, Mar 06 2019

Formula

a(n) = n^2 - binomial(n+1, n-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
a(n) = binomial(n,2)+binomial(n,1)-binomial(n,3). - Zerinvary Lajos, Jul 23 2006
a(n) = n*(1+6*n-n^2)/6. - Karen A. Yeats, Nov 20 2006
From Michael Somos, Jul 04 2012: (Start)
G.f.: x * (1 - x - x^2) / (1 - x)^4.
a(-1 - n) = A008778(n). (End)
E.g.f.: x*(6 +3*x -x^2)*exp(x)/6. - G. C. Greubel, Mar 06 2019

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
Offset changed to 1 at the suggestion of Michel Marcus, Mar 05 2019
Formulas and programs addapted for offset 1 by Michel Marcus, Mar 05 2019

A081499 Sum at 45 degrees to horizontal in triangle of A081498.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 12, 16, 15, 20, 16, 22, 14, 21, 8, 16, -3, 6, -20, -10, -44, -33, -76, -64, -117, -104, -168, -154, -230, -215, -304, -288, -391, -374, -492, -474, -608, -589, -740, -720, -889, -868, -1056, -1034, -1242, -1219, -1448, -1424, -1675, -1650, -1924, -1898, -2196, -2169, -2492, -2464, -2813
Offset: 1

Views

Author

Amarnath Murthy, Mar 25 2003

Keywords

Comments

The leading diagonal is given by A080956(n) = ((n+1)(2-n)/2).

Examples

			a(7) = 7+5+2+(-2) = 12.

Links

Crossrefs

Cf. A080956, A081498.

Programs

  • Maple
    seq((n+floor(n/2)+1)*(n-floor(n/2))/2-binomial(ceil(n/2)+1,ceil(n/2)-2),n=1..60); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,2,4,6,8,11,12},60] (* Harvey P. Dale, Jan 17 2022 *)
  • PARI
    Vec(x*(1 + x - x^2 - x^3 - x^4) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ Colin Barker, Nov 12 2017

Formula

a(n) = (n+floor(n/2)+1)*(n-floor(n/2))/2-binomial(ceiling(n/2)+1, ceiling(n/2)-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
G.f.: x*(1 + x - x^2 - x^3 - x^4) / ((1 - x)^4*(1 + x)^3). - Colin Barker, Dec 18 2012
From Colin Barker, Nov 12 2017: (Start)
a(n) = (1/96)*(-2*n^3 + 36*n^2 + 32*n) for n even.
a(n) = (1/96)*(-2*n^3 + 30*n^2 + 50*n + 18) for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7.
(End)

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

A202480 Riordan array (1/(1-x), x(2x-1)/(1-x)^2).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, 0, 1, -1, 1, 2, -1, -1, 1, 1, 5, -5, 2, 1, -1, 1, 9, -10, 8, -3, -1, 1, 1, 14, -14, 14, -11, 4, 1, -1, 1, 20, -14, 14, -17, 14, -5, -1, 1, 1, 27, -6, 0, -9, 19, -17, 6, 1, -1
Offset: 0

Views

Author

Philippe Deléham, Dec 20 2011

Keywords

Comments

Row sums are Fibonacci(n-1) = A000045(n-1).
Diagonal sums are A078003(n).
(Sum_{j, 0<=j<=k} T(k,j))/(1-2x)^k gives g.f. of column A165241(n+k-1,k-1) in triangular array in A165241.

Examples

			Triangle begins :
1
1, -1
1, -1, 1
1, 0, 1, -1
1, 2, -1, -1, 1
1, 5, -5, 2, 1, -1
1, 9, -10, 8, -3, -1, 1
1, 14, -14, 14, -11, 4, 1, -1
(1+x^2-x^3)/(1-2x)^3 is the g.f of column A165241(n+2,2) := 1, 6, 25, 85, 258, 728, 1952, 5040, ...

Crossrefs

Cf. A000012, A080956, A000045, A078003, A124341, A165241

Formula

T(n,k) = 2*T(n-1,k) + 2*T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
T(n,k) = (-1)^n*A124341(n,k).

A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10
Offset: 0

Views

Author

Peter Luschny, Aug 04 2019

Keywords

Comments

A formal extension of the figurative numbers A139600 to negative n.

Examples

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Links

Crossrefs

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

Programs

  • Maple
    A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);
  • Python
    def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A326815 Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 2 * p^(-s)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, -2, 0, 1, 1, 0, 1, 1, 1, -5, 1, 0, 1, 0, 1, 1, 1, -2, 0, 1, -2, 0, 1, 1, 1, -9, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 0, 1, 1, -5, 0, 0, 1, 0, 1, -2, 1, -2, 1, 1, 1, 0, 1, 1, 0, -14, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -5, -5, 1, 1, 0, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 19 2019

Keywords

Comments

Inverse Moebius transform applied twice to A076479 (unitary Moebius function).

Links

Crossrefs

Cf. A000005, A001221, A005117 (positions of 1's), A007425, A008683, A038109 (positions of 0's), A046951, A076479, A080956, A326814.

Programs

  • Mathematica
    Table[Sum[(-1)^PrimeNu[n/d] DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := (e + 1)*(2 - e)/2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)
  • PARI
    A326815(n) = sumdiv(n,d,((-1)^omega(n/d))*numdiv(d)); \\ Antti Karttunen, Nov 17 2019
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

Formula

a(n) = Sum_{d|n} (-1)^omega(n/d) * tau(d), where omega = A001221 and tau = A000005.
a(n) = Sum_{d|n} tau_3(n/d) * mu(d) * 2^omega(d), where tau_3 = A007425 and mu = A008683.
Multiplicative with a(p^e) = (e+1)*(2-e)/2 = A080956(e). - Amiram Eldar, Oct 26 2020

A129685 Exponential Riordan array [1-x^2/2, x].

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 0, 0, -6, 0, 1, 0, 0, 0, -10, 0, 1, 0, 0, 0, 0, -15, 0, 1, 0, 0, 0, 0, 0, -21, 0, 1, 0, 0, 0, 0, 0, 0, -28, 0, 1, 0, 0, 0, 0, 0, 0, 0, -36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -45, 0, 1
Offset: 0

Views

Author

Paul Barry, Apr 28 2007

Keywords

Examples

			Triangle begins
1,
0, 1,
-1, 0, 1,
0, -3, 0, 1,
0, 0, -6, 0, 1,
0, 0, 0, -10, 0, 1,
0, 0, 0, 0, -15, 0, 1,
0, 0, 0, 0, 0, -21, 0, 1,
0, 0, 0, 0, 0, 0, -28, 0, 1,
0, 0, 0, 0, 0, 0, 0, -36, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, -45, 0, 1

Crossrefs

Inverse of A129684. Row sums are A080956.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[1 - #^2/2&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

A155726 Production matrix for Fibonacci numbers, read by row.

Original entry on oeis.org

0, 1, 2, -1, 1, 3, 0, -1, 1, 4, 0, 0, -1, 1, 5, 0, 0, 0, -1, 1, 6, 0, 0, 0, 0, -1, 1, 7, 0, 0, 0, 0, 0, -1, 1, 8, 0, 0, 0, 0, 0, 0, -1, 1, 9, 0, 0, 0, 0, 0, 0, 0, -1
Offset: 0

Views

Author

Paul Barry, Jan 25 2009

Keywords

Comments

The matrix generated by this matrix has row sums F(n+1).

Examples

			Matrix begins
  0, 1,
  2, -1, 1,
  3, 0, -1, 1,
  4, 0, 0, -1, 1,
  5, 0, 0, 0, -1, 1,
  6, 0, 0, 0, 0, -1, 1,
  7, 0, 0, 0, 0, 0, -1, 1,
  8, 0, 0, 0, 0, 0, 0, -1, 1,
  9, 0, 0, 0, 0, 0, 0, 0, -1, 1
The row augmented triangular matrix
  1,
  0, 1,
  2, -1, 1,
  3, 0, -1, 1,
  4, 0, 0, -1, 1,
  5, 0, 0, 0, -1, 1,
  6, 0, 0, 0, 0, -1, 1,
  7, 0, 0, 0, 0, 0, -1, 1,
  8, 0, 0, 0, 0, 0, 0, -1, 1,
  9, 0, 0, 0, 0, 0, 0, 0, -1, 1
has row sums 0^n+n. Its inverse has row sums (n+1)(2-n)/2 or A080956.
This is the matrix
    1,
    0, 1,
   -2, 1, 1,
   -5, 1, 1, 1,
   -9, 1, 1, 1, 1,
  -14, 1, 1, 1, 1, 1,
  -20, 1, 1, 1, 1, 1, 1,
  -27, 1, 1, 1, 1, 1, 1, 1,
  -35, 1, 1, 1, 1, 1, 1, 1, 1
with first column (n+2)(1-n)/2.

Links

  • E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Crossrefs

Cf. A080956, A155727.

A159856 Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.

Original entry on oeis.org

1, 2, -1, 3, -4, 1, 4, -11, 6, -1, 5, -26, 23, -8, 1, 6, -57, 72, -39, 10, -1, 7, -120, 201, -150, 59, -12, 1, 8, -247, 522, -501, 268, -83, 14, -1, 9, -502, 1291, -1524, 1037, -434, 111, -16, 1, 10, -1013, 3084, -4339, 3598, -1905, 656, -143, 18, -1
Offset: 0

Views

Author

Philippe Deléham, Apr 24 2009

Keywords

Comments

A Riordan array - see the Luzon references.
The second column is A000295 signed. - Michel Marcus, Feb 14 2014

Examples

			Triangle begins
  1;
  2,   -1;
  3,   -4,    1;
  4,  -11,    6,   -1;
  5,  -26,   23,   -8,    1;
  6,  -57,   72,  -39,   10,   -1;
  7, -120,  201, -150,   59,  -12,    1;
  ...

Links

Crossrefs

Cf. A000295, A047926, A080956, A181690.

Programs

  • Mathematica
    With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
+y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)
  • Maxima
    T(n,k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k,x,0,15),x,n); /* Vladimir Kruchinin, Nov 22 2016 */

Formula

From R. J. Mathar, May 31 2009: (Start)
Sum_{k=0..n} T(n,k) = A080956(n).
Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k-1) for k > 0. - Vladimir Kruchinin, Nov 22 2016 [corrected by Werner Schulte, May 09 2024]
G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - Vladimir Kruchinin, Nov 22 2016

Extensions

a(41) corrected by Georg Fischer, Feb 18 2020

A175631 a(n) = (n-th pentagonal number) modulo (n-th triangular number).

Original entry on oeis.org

0, 2, 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430
Offset: 1

Views

Author

Zak Seidov, Jul 29 2010

Keywords

Examples

			a(1)=0 because (1(3-1)/2) mod (1(1+1)/2) = 1 mod 1 = 0,
a(2)=2 because (2(6-1)/2) mod (2(2+1)/2) = 5 mod 3 = 2.

Links

Crossrefs

Cf. A000096 (n(n+3)/2), A000217 (triangular numbers), A000326 (pentagonal numbers), A175630 (n-th pentagonal number mod (n+2)).
Cf. A080956, A132315, A132316, A132317.

Programs

  • Magma
    [n lt 4 select 1+(-1)^n else n*(n-3)/2: n in [1..60]]; // G. C. Greubel, Jan 30 2022
  • Mathematica
    Table[Mod[n(3n-1)/2, n(n+1)/2],{n,100}]
Module[{nn=60},Mod[#[[1]],#[[2]]]&/@Thread[{PolygonalNumber[ 5,Range[ nn]],Accumulate[ Range[nn]]}]] (* Harvey P. Dale, Nov 19 2022 *)
  • Sage
    def A175631(n): return 1+(-1)^n if (n<4) else 9*binomial(n/3, 2)
[A175631(n) for n in (1..60)] # G. C. Greubel, Jan 30 2022

Formula

For n>=3, a(n) = A000096(n-2).
From Chai Wah Wu, Oct 12 2018: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^2*(2 - 6*x + 8*x^2 - 3*x^3)/(1 - x)^3. (End)
E.g.f.: (x/2)*(2 + 3*x - (2 - x)*exp(x)). - G. C. Greubel, Jan 30 2022
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