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 1-5 of 5 results.

A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 2, -2, -3, 4, 5, 0, -5, -4, 5, 9, 5, -5, -9, -5, 6, 14, 14, 0, -14, -14, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 12 2012

Keywords

Examples

			The triangle begins:
    0:                              0
    1:                            1   -1
    2:                          2   0   -2
    3:                       3    2   -2   -3
    4:                     4    5   0   -5   -4
    5:                  5    9    5   -5   -9   -5
    6:                6   14   14   0  -14  -14   -6
    7:             7   20   28   14  -14  -28  -20   -7
    8:           8   27   48   42   0  -42  -48  -27   -8
    9:        9   35   75   90   42  -42  -90  -75  -35   -9
   10:     10   44  110  165  132   0 -132 -165 -110  -44  -10
   11:  11   54  154  275  297  132 -132 -297 -275 -154  -54  -11  .

Links

Crossrefs

Cf. A007318, A000004, A000096, A000108, A000245, A000344, A000588, A000589, A000590, A001392, A002057, A003517, A003518, A003519, A005557, A005586, A005587, A008313, A014495, A064059, A064061, A080956, A090749, A097808, A112467, A124087, A124088, A129936, A259525.

Programs

  • Haskell
    a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
   where diff row = zipWith (-) (tail row) row
  • Mathematica
    row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)

Formula

T(n,k) = A007318(n+1,k+1) - A007318(n+1,k), 0<=k<=n, i.e. first differences of rows in Pascal's triangle;
T(n,k) = -T(n,k);
row sums and central terms equal 0, cf. A000004;
sum of positive elements of n-th row = A014495(n+1);
T(n,0) = n;
T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;
T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);
T(n,3) = A005587(n-6) for n > 5;
T(n,4) = A005557(n-9) for n > 8;
T(n,5) = A064059(n-11) for n > 10;
T(n,6) = A064061(n-13) for n > 12;
T(n,7) = A124087(n) for n > 14;
T(n,8) = A124088(n) for n > 16;
T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;
T(2*n+3,n) = A000245(n+2);
T(2*n+4,n) = A002057(n+1);
T(2*n+5,n) = A000344(n+3);
T(2*n+6,n) = A003517(n+3);
T(2*n+7,n) = A000588(n+4);
T(2*n+8,n) = A003518(n+4);
T(2*n+9,n) = A001392(n+5);
T(2*n+10,n) = A003519(n+5);
T(2*n+11,n) = A000589(n+6);
T(2*n+12,n) = A090749(n+6);
T(2*n+13,n) = A000590(n+7).

A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

Original entry on oeis.org

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0
Offset: 0

Views

Author

Paul Curtz, Jun 24 2011

Keywords

Comments

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1, -1, -1, -1,
1, 0, -1, -2, -3, -4, -5, -6,
1, 1, 0, -2, -5, -9, -14, -20,
1, 2, 2, 0, -5, -14, -28, -48,
1, 3, 5, 5, 0, -14, -42, -90,
1, 4, 9, 14, 14, 0, -42, -132,
1, 5, 14, 28, 42, 42, 0, -132,
1, 6, 20, 48, 90, 132, 132, 0.
The rows in this array are essentially the columns of T(n,k).

Examples

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

Crossrefs

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.
Cf. A194084. - Paul Curtz, Aug 16 2011

Programs

  • Maple
    p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Formula

Sum_{k=0..n} T(n,k) = A057079(n-1).
Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

A176239 Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.

Original entry on oeis.org

0, -1, 1, -1, 0, 2, 0, 1, -2, 2, 0, -5, 0, 0, 1, -3, 5, -5, 0, 14, 0, 0, 0, 1, -4, 9, -14, 14, 0, -42, 0, 0, 0, 0, 1, -5, 14, -28, 42, -42, 0, 132, 0, 0, 0, 0, 0, 1, -6, 20, -48, 90, -132, 132, 0, -429, 0, 0, 0, 0, 0, 0, 1, -7, 27, -75, 165, -297, 429, -429, 0, 1430
Offset: 0

Views

Author

Paul Curtz, Apr 12 2010

Keywords

Examples

			The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
0,-1;                          (-1)^k*k  A001477
1,-1,.0,.2;                      (-1)^(k+1)*(k+1)*(k-2)/2  A080956, A000096
0,.1,-2,.2,.0,-5;                 (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
0,.0,.1,-3,.5,-5,..0,.14;             (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
0,.0,.0,.1,-4,.9,-14,.14,.0,-42;           A005557, A034807
0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;

Crossrefs

Cf A140344, A033184.

Programs

  • Maple
    A009766 := proc(n,k) if k<0 or k >n then 0; else binomial(n+k,n)*(n-k+1)/(n+1) ; end if; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A176239 := proc(n,k) if k <= 2*n-1 then (-1)^(n+k+1)*A009766(n,k-n+1) elif k = 2*n then 0; elif k < 2*(n+1) then (-1)^(n+1)*A000108(n+1); else 0; end if; end proc: # R. J. Mathar, Dec 03 2010

Formula

T(n,k) = T(n+1,k)+T(n+1,k+1), k <= 2n+1.
T(n,2n) = 0.
T(n,2n+1) = (-1)^(n+1)*A000108(n+1).
T(n,k) = (-1)^(n+k+1)*A009766(n,k-n+1), k < 2n.

A254749 1-gonal pyramidal numbers.

Original entry on oeis.org

1, 2, 2, 0, -5, -14, -28, -48, -75, -110, -154, -208, -273, -350, -440, -544, -663, -798, -950, -1120, -1309, -1518, -1748, -2000, -2275, -2574, -2898, -3248, -3625, -4030, -4464, -4928, -5423, -5950, -6510, -7104, -7733, -8398, -9100, -9840, -10619, -11438
Offset: 1

Views

Author

Colin Barker, Feb 07 2015

Keywords

Comments

Not strictly pyramidal numbers, but the result of using the Wikipedia formula with r = 1.
Essentially the same as A129936 and A005586.

Examples

			G.f. = x + 2*x^2 + 2*x^3 - 5*x^5 - 14*x^6 - 28*x^7 - 48*x^8 - 75*x^9 + ...

Links

Crossrefs

Cf. A005581, A005586, A129936.

Programs

  • Magma
    [(n*(4+3*n-n^2))/6: n in [1..60]]; // G. C. Greubel, Aug 03 2018
  • Mathematica
    Table[(n*(4+3*n-n^2))/6, {n,1,60}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1, 2, 2, 0}, 60] (* G. C. Greubel, Aug 03 2018 *)
  • PARI
    ppg(r, n) = (3*n^2+n^3*(r-2)-n*(r-5))/6
vector(100, n, ppg(1, n))

Formula

a(n) = n*(4 + 3*n - n^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(1 - 2*x)/(x-1)^4.
a(n) = A005581(-n) = -A005586(n-4) = -A129936(n-2) for all n in Z. - Michael Somos, Jul 28 2015
E.g.f.: -exp(x)*x*(x^2 - 6)/6. - Elmo R. Oliveira, Aug 04 2025

A212346 Sequence of coefficients of x^0 in marked mesh pattern generating function Q_{n,132}^(0,4,0,0)(x).

Original entry on oeis.org

1, 1, 2, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733
Offset: 0

Views

Author

N. J. A. Sloane, May 09 2012

Keywords

Comments

Conjecture stated in Formula holds through a(35).

Links

Programs

  • Mathematica
    QQ0[t, x] = (1 - (1-4*x*t)^(1/2)) / (2*x*t); QQ1[t, x] = 1/(1 - t*QQ0[t, x]); QQ2[t, x] = (1 + t*(QQ1[t, x] - QQ0[t, x]))/(1 - t*QQ0[t, x]); QQ3[t, x] = (1 + t*(QQ2[t, x] - QQ0[t, x] + t*(QQ1[t, x] - QQ0[t,  x])))/(1 - t*QQ0[t, x]); QQ4[t, x] = (1 + t*(QQ3[t, x] - QQ0[t, x] + t*(QQ2[t, x] - QQ0[t, x]) + (2*t^2*(QQ1[t, x] - QQ0[t, x]))))/(1 - t*QQ0[t, x]); q=Simplify[Series[QQ4[t, x], {t, 0, 35}]]; CoefficientList[q /. x -> 0, t]  (* Robert Price, Jun 04 2012 *)

Formula

Conjecture: this appears to equal (n+3)(n^2-4)/6 for n >= 3, see A129936.
Conjectures from Colin Barker, Feb 11 2015: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>6.
G.f.: (2*x^6-5*x^5+3*x^4-x^3+4*x^2-3*x+1) / (x-1)^4.
(End)

Extensions

a(10)-a(35) from Robert Price, Jun 02 2012
Showing 1-5 of 5 results.

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

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