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 1-3 of 3 results.

A093561 (4,1) Pascal triangle.

Original entry on oeis.org

1, 4, 1, 4, 5, 1, 4, 9, 6, 1, 4, 13, 15, 7, 1, 4, 17, 28, 22, 8, 1, 4, 21, 45, 50, 30, 9, 1, 4, 25, 66, 95, 80, 39, 10, 1, 4, 29, 91, 161, 175, 119, 49, 11, 1, 4, 33, 120, 252, 336, 294, 168, 60, 12, 1, 4, 37, 153, 372, 588, 630, 462, 228, 72, 13, 1, 4, 41, 190, 525, 960, 1218
Offset: 0

Views

Author

Wolfdieter Lang, Apr 22 2004

Keywords

Comments

The array F(4;n,m) gives in the columns m >= 1 the figurate numbers based on A016813, including the hexagonal numbers A000384 (see the W. Lang link).
This is the fourth member, d=4, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 and A093560, for d=1..3.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+3*z)/(1-(1+x)*z).
The SW-NE diagonals give A000285(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018

Examples

			Triangle begins
  [1];
  [4, 1];
  [4, 5, 1];
  [4, 9, 6, 1];
  ...

References

  • Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
  • Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Links

Crossrefs

Cf. Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 3 for n=2 and 0 otherwise.
Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, A051947, A050483, A052181, A055843.
Cf. A007318, A093562 (d=5), A228196, A228576.

Programs

  • Haskell
    a093561 n k = a093561_tabl !! n !! k
a093561_row n = a093561_tabl !! n
a093561_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [4, 1]
-- Reinhard Zumkeller, Aug 31 2014
  • Python
    from math import comb, isqrt
def A093561(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+3*(r-a))//r if n else 1 # Chai Wah Wu, Nov 12 2024

Formula

a(n, m) = F(4;n-m, m) for 0<= m <= n, otherwise 0, with F(4;0, 0)=1, F(4;n, 0)=4 if n>=1 and F(4;n, m) = (4*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 3*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 9*x + 6*x^2/2! + x^3/3!) = 4 + 13*x + 28*x^2/2! + 50*x^3/3! + 80*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A055843 Expansion of (1+3*x)/(1-x)^10.

Original entry on oeis.org

1, 13, 85, 385, 1375, 4147, 11011, 26455, 58630, 121550, 238238, 445094, 797810, 1379210, 2309450, 3759074, 5965487, 9253475, 14060475, 20967375, 30735705, 44352165, 63081525, 88529025, 122713500, 168152556, 227961228, 305965660, 406833460, 536222500, 700950052
Offset: 0

Views

Author

Barry E. Williams, May 30 2000

Keywords

Comments

Partial sums of A052181.

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A052181.
Cf. A093561 ((4, 1) Pascal, column m=9).

Programs

  • GAP
    List([0..30], n-> (4*n+9)*Binomial(n+8, 8)/9 ); # G. C. Greubel, Jan 21 2020
  • Magma
    [(4*n+9)*Binomial(n+8, 8)/9: n in [0..30]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( (4*n+9)*binomial(n+8, 8)/9, n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[4*Binomial[n+9,9] - 3*Binomial[n+8,8], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    vector(31, n, (4*n+5)*binomial(n+7, 8)/9) \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(4*n+9)*binomial(n+8, 8)/9 for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (4*n+9)*binomial(n+8, 8)/9.
G.f.: (1+3*x)/(1-x)^10.
a(n) = 4*binomial(n+9,9) - 3*binomial(n+8,8). - G. C. Greubel, Jan 21 2020
Sum_{n>=0} 1/a(n) = 9437184*Pi/24035 + 56623104*log(2)/24035 - 482087736/168245. - Amiram Eldar, Feb 17 2023

A027819 a(n) = 7*(n+1)*binomial(n+6,7)/2.

Original entry on oeis.org

7, 84, 504, 2100, 6930, 19404, 48048, 108108, 225225, 440440, 816816, 1447992, 2469012, 4069800, 6511680, 10147368, 15444891, 23015916, 33649000, 48348300, 68378310, 95315220, 131105520, 178132500, 239291325, 318073392, 418660704, 546031024, 706074600, 905723280
Offset: 1

Views

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

Comments

Number of 14-subsequences of [ 1, n ] with just 6 contiguous pairs.

Links

Crossrefs

Cf. A052181.

Programs

  • Mathematica
    Table[7*(n + 1)*Binomial[n + 6, 7]/2, {n, 1, 50}] (* Amiram Eldar, Feb 03 2022 *)

Formula

a(n) = 7*A052181(n).
G.f.: 7*(1+3*x)*x/(1-x)^9.
a(n) = C(n+1,2)*C(n+6,6). - Zerinvary Lajos, May 26 2005
From Amiram Eldar, Feb 03 2022: (Start)
Sum_{n>=1} 1/a(n) = 5969/300 - 2*Pi^2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 384*log(2)/5 + 13049/300. (End)
Showing 1-3 of 3 results.

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