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 1-10 of 26 results. Next

A274180 a(n) = Sum_{k=0..4n} (A035343(n,k) mod 2) * 2^k.

Original entry on oeis.org

1, 31, 341, 6483, 69905, 2027247, 21041413, 417263459, 4311810305, 133666119455, 1461703693397, 27806864656979, 299071474565137, 8708265758097903, 90161415181374469, 1785159701350222947, 18447025552981295105
Offset: 0

Views

Author

Gheorghe Coserea, Jun 12 2016

Keywords

Comments

a(n) is a binary palindrome (A006995) of 4n+1 bits since A035343(n,k) = A035343(n,4n-k), k=0..4n and A035343(n,0) = A035343(n,4n) = 1.

Links

Crossrefs

Cf. A035343.

Programs

  • PARI
    a(n) =  subst(lift(Pol(Mod([1, 1, 1, 1, 1], 2), 'x)^n), 'x, 2);
vector(17,n,a(n-1))

A000012 The simplest sequence of positive numbers: the all 1's sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, May 16 1994

Keywords

Comments

Number of ways of writing n as a product of primes.
Number of ways of writing n as a sum of distinct powers of 2.
Continued fraction for golden ratio A001622.
Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner, Sep 08 2002
An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - Don Reble, Apr 17 2005
Binomial transform of A000007; inverse binomial transform of A000079. - Philippe Deléham, Jul 07 2005
A063524(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2008
For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). - K.V.Iyer, Apr 11 2009
The partial sums give the natural numbers (A000027). - Daniel Forgues, May 08 2009
From Enrique Pérez Herrero, Sep 04 2009: (Start)
a(n) is also tau_1(n) where tau_2(n) is A000005.
a(n) is a completely multiplicative arithmetical function.
a(n) is both squarefree and a perfect square. See A005117 and A000290. (End)
Also smallest divisor of n. - Juri-Stepan Gerasimov, Sep 07 2009
Also decimal expansion of 1/9. - Enrique Pérez Herrero, Sep 18 2009; corrected by Klaus Brockhaus, Apr 02 2010
a(n) is also the number of complete graphs on n nodes. - Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009
Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. - Jaroslav Krizek, Oct 18 2009
n-th prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th noncomposite number. - Juri-Stepan Gerasimov, Oct 26 2009
For all n>0, the sequence of limit values for a(n) = n!*Sum_{k>=n} k/(k+1)!. Also, a(n) = n^0. - Harlan J. Brothers, Nov 01 2009
a(n) is also the number of 0-regular graphs on n vertices. - Jason Kimberley, Nov 07 2009
Differences between consecutive n. - Juri-Stepan Gerasimov, Dec 05 2009
From Matthew Vandermast, Oct 31 2010: (Start)
1) When sequence is read as a regular triangular array, T(n,k) is the coefficient of the k-th power in the expansion of (x^(n+1)-1)/(x-1).
2) Sequence can also be read as a uninomial array with rows of length 1, analogous to arrays of binomial, trinomial, etc., coefficients. In a q-nomial array, T(n,k) is the coefficient of the k-th power in the expansion of ((x^q -1)/(x-1))^n, and row n has a sum of q^n and a length of (q-1)*n + 1. (End)
The number of maximal self-avoiding walks from the NW to SW corners of a 2 X n grid.
When considered as a rectangular array, A000012 is a member of the chain of accumulation arrays that includes the multiplication table A003991 of the positive integers. The chain is ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.) - Clark Kimberling, Feb 06 2011
a(n) = A007310(n+1) (Modd 3) := A193680(A007310(n+1)), n>=0. For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the three residue classes Modd 3, called [0], [1], and [2], are shown in the array A088520, if there the third row is taken as class [0] after inclusion of 0. - Wolfdieter Lang, Feb 09 2012
Let M = Pascal's triangle without 1's (A014410) and V = a variant of the Bernoulli numbers A027641 but starting [1/2, 1/6, 0, -1/30, ...]. Then M*V = [1, 1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
As a lower triangular array, T is an example of the fundamental generalized factorial matrices of A133314. Multiplying each n-th diagonal by t^n gives M(t) = I/(I-t*S) = I + t*S + (t*S)^2 + ... where S is the shift operator A129184, and T = M(1). The inverse of M(t) is obtained by multiplying the first subdiagonal of T by -t and the other subdiagonals by zero, so A167374 is the inverse of T. Multiplying by t^n/n! gives exp(t*S) with inverse exp(-t*S). - Tom Copeland, Nov 10 2012
The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole. According to that historical definition, the length of one degree of latitude, that is, 60 nautical miles, would be exactly 111111.111... meters. - Jean-François Alcover, Jun 02 2013
Deficiency of 2^n. - Omar E. Pol, Jan 30 2014
Consider n >= 1 nonintersecting spheres each with surface area S. Define point p on sphere S_i to be a "public point" if and only if there exists a point q on sphere S_j, j != i, such that line segment pq INTERSECT S_i = {p} and pq INTERSECT S_j = {q}; otherwise, p is a "private point". The total surface area composed of exactly all private points on all n spheres is a(n)*S = S. ("The Private Planets Problem" in Zeitz.) - Rick L. Shepherd, May 29 2014
For n>0, digital roots of centered 9-gonal numbers (A060544). - Colin Barker, Jan 30 2015
Product of nonzero digits in base-2 representation of n. - Franklin T. Adams-Watters, May 16 2016
Alternating row sums of triangle A104684. - Wolfdieter Lang, Sep 11 2016
A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016
Length of period of continued fraction for sqrt(A002522) or sqrt(A002496). - A.H.M. Smeets, Oct 10 2017
a(n) is also the determinant of the (n+1) X (n+1) matrix M defined by M(i,j) = binomial(i,j) for 0 <= i,j <= n, since M is a lower triangular matrix with main diagonal all 1's. - Jianing Song, Jul 17 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j) for 1 <= i,j <= n (see Xavier Merlin reference). - Bernard Schott, Dec 05 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = tau(gcd(i,j)) for 1 <= i,j <= n (see De Koninck & Mercier reference). - Bernard Schott, Dec 08 2020

Examples

			1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) = A001622.
1/9 = 0.11111111111111...
From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 7 for nonnegative odd numbers not divisible by 3:
A007310: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
Modd 3:  1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
(End)

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
  • J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 692 pp. 90 and 297, Ellipses, Paris, 2004.
  • Xavier Merlin, Méthodix Algèbre, Exercice 1-a), page 153, Ellipses, Paris, 1995.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 277, 284.
  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
  • Paul Zeitz, The Art and Craft of Mathematical Problem Solving, The Great Courses, The Teaching Company, 2010 (DVDs and Course Guidebook, Lecture 6: "Pictures, Recasting, and Points of View", pp. 32-34).

Links

Crossrefs

Cf. A000004, A007395, A010701, A000027, A027641, A014410, A211216, A212393, A060544, A051801, A104684.
For other q-nomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890. - Matthew Vandermast, Oct 31 2010
Cf. A097805, A118800, A130595, A167374, A008284 (multisets).

Programs

  • Haskell
    a000012 = const 1
a000012_list = repeat 1 -- Reinhard Zumkeller, May 07 2012
  • Magma
    [1 : n in [0..100]];
  • Maple
    seq(1, i=0..150);
  • Mathematica
    Array[1 &, 50] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
  • Maxima
    makelist(1, n, 1, 30); /* Martin Ettl, Nov 07 2012 */
  • PARI
    {a(n) = 1};
  • Python
    print([1 for n in range(90)]) # Michael S. Branicky, Apr 04 2022

Formula

a(n) = 1.
G.f.: 1/(1-x).
E.g.f.: exp(x).
G.f.: Product_{k>=0} (1 + x^(2^k)). - Zak Seidov, Apr 06 2007
Completely multiplicative with a(p^e) = 1.
Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f. Sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. Sum T(n,m) x^n y^m/m! = e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)), e.g.f. Sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - Franklin T. Adams-Watters, Feb 06 2006
Dirichlet g.f.: zeta(s). - Ilya Gutkovskiy, Aug 31 2016
a(n) = Sum_{l=1..n} (-1)^(l+1)*2*cos(Pi*l/(2*n+1)) = 1 identically in n >= 1 (for n=0 one has 0 from the undefined sum). From the Jolley reference, (429) p. 80. Interpretation: consider the n segments between x=0 and the n positive zeros of the Chebyshev polynomials S(2*n, x) (see A049310). Then the sum of the lengths of every other segment starting with the one ending in the largest zero (going from the right to the left) is 1. - Wolfdieter Lang, Sep 01 2016
As a lower triangular matrix, T = M*T^(-1)*M = M*A167374*M, where M(n,k) = (-1)^n A130595(n,k). Note that M = M^(-1). Cf. A118800 and A097805. - Tom Copeland, Nov 15 2016

A001591 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5), a(0)=a(1)=a(2)=a(3)=0, a(4)=1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061, 1333610936, 2621810068
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of permutations satisfying -k <= p(i) - i <= r, i=1..n-4, with k=1, r=4. - Vladimir Baltic, Jan 17 2005
a(n) is the number of compositions of n-4 with no part greater than 5. - Vladimir Baltic, Jan 17 2005
The pentanomial (A035343(n)) transform of a(n) is a(5n+4), n >= 0. - Bob Selcoe, Jun 10 2014
a(n) is the number of ways to tile a strip of length n-4 with squares, dominoes, trominoes (of length 3), and rectangles with length 4 (tetraminoes) and length 5 (pentaminoes). - Wajdi Maaloul, Jun 21 2022

Examples

			n=2: a(14) = (1*1 + 2*1 + 3*2 + 4*4 + 5*8 + 4*16 + 3*31 + 2*61 + 1*120) = 464. - _Bob Selcoe_, Jun 10 2014
G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 31*x^10 + 120*x^11 + ...

References

  • Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Cf. A106303 (Pisano period lengths).
Cf. A035343 (pentanomial coefficients).
Cf. A074048, A123127, A123126, A074062.

Programs

  • Magma
    a:=[0,0,0,0,1]; [n le 5 select a[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..40]]; // Marius A. Burtea, Oct 03 2019
  • Maple
    g:=1/(1-z-z^2-z^3-z^4-z^5): gser:=series(g, z=0, 49): seq((coeff(gser, z, n)), n=-4..32); # Zerinvary Lajos, Apr 17 2009
# second Maple program:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>, <1|1|1|1|1>>^n)[1, 5]:
seq(a(n), n=0..44);  # Alois P. Heinz, Apr 09 2021
  • Mathematica
    CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], x]
a[0] = a[1] = a[2] = a[3] = 0; a[4] = a[5] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 6]; Array[a, 37, 0]
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
  • Maxima
    a(n):=mod(floor(10^((n-4)*(n+1))*10^(5*(n+1))*(10^(n+1)-1)/(10^(6*(n+1))-2*10^(5*(n+1))+1)),10^n); /* Tani Akinari, Apr 10 2014 */
  • PARI
    a=vector(100);a[4]=a[5]=1;for(n=6,#a,a[n]=a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]);concat(0, a) \\ Charles R Greathouse IV, Jul 15 2011
  • PARI
    A001591(n,m=5)=(matrix(m,m,i,j,i==j-1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 20 2018
  • PARI
    a(n)= {my(x='x, p=polrecip(1 - x - x^2 - x^3 - x^4 - x^5)); polcoef(lift(Mod(x, p)^n), 4); }
vector(41, n, a(n-1)) \\ Joerg Arndt, May 16 2021
  • Python
    def pentanacci():
    a, b, c, d, e = 0, 0, 0, 0, 1
    while True:
        yield a
        a, b, c, d, e = b, c, d, e, a + b + c + d + e
f = pentanacci()
print([next(f) for  in range(100)]) # _Reza K Ghazi Apr 09 2021

Formula

G.f.: x^4/(1 - x - x^2 - x^3 - x^4 - x^5). - Simon Plouffe in his 1992 dissertation.
G.f.: Sum_{n >= 0} x^(n+4) * (Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + x^4)/(1 + k*x + k*x^2 + k*x^3 + k*x^4)). - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^4-z^5)/(1-2*z+z^6); then a(n) = Sum_{i=0..floor((n-4)/6)} ((-1)^i*binomial(n-4-5*i,i)*2^(n-4-6*i)) - Sum_{i=0..floor((n-5)/6)} ((-1)^i*binomial(n-5-5*i,i)*2^(n-5-6*i)) with convention Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n) = Sum_{k=1..n} (Sum_{r=0..k} (binomial(k,r) * Sum_{m=0..r} (binomial(r,m) * Sum_{j=0..m} (binomial(m,j)*binomial(j,n-m-k-j-r))))), n > 0. - Vladimir Kruchinin, Aug 30 2010
Sum_{k=0..4*n} a(k+b)*A035343(n,k) = a(5*n+b), b >= 0.
a(n) = 2*a(n-1) - a(n-6). - Vincenzo Librandi, Dec 19 2010
a(n) = (Sum_{i=0..n-1} a(i)*A074048(n-i))/(n-4) for n > 4. - Greg Dresden and Advika Srivastava, Oct 01 2019
For k>0 and n>0, a(n+5*k) = A074048(k)*a(n+4*k) - A123127(k-1)*a(n+3*k) + A123126(k-1)*a(n+2*k) - A074062(k)*a(n+k) + a(n). - Kai Wang, Sep 06 2020
lim n->oo a(n)/a(n-1) = A103814. - R. J. Mathar, Mar 11 2024

A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n.

Original entry on oeis.org

1, 1, 5, 19, 85, 381, 1751, 8135, 38165, 180325, 856945, 4091495, 19611175, 94309099, 454805755, 2198649549, 10651488789, 51698642405, 251345549849, 1223798004815, 5966636799745, 29125608152345, 142330448514875, 696235630761115, 3408895901222375
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficient of x^n in ((1-x^10)/((1-x^5)(1-x^2)(1-x)))^n. - Michael Somos, Sep 24 2003
Note that n divides a(n+1) - a(n). - T. D. Noe, Mar 16 2005
Terms that are not a multiple of 5 have zero density, namely, there are fewer than n^(log(4)/log(5)) such terms among A005191(1..n). In particular, A005191(5k+2) and A005191(5k+4) are multiples of 5 for every k. - Max Alekseyev, Apr 25 2005
Number of n-step 1-D walks ending at the origin with steps of size 0, 1 or 2. - David Scambler, Apr 09 2012
Number of compositions of 2n into exactly n nonnegative parts <= four. a(2) = 5: [4,0], [3,1], [2,2], [1,3], [0,4]. - Alois P. Heinz, Sep 13 2018
Let f(m) = ceiling((q+log(q))/log(25)), where q = -log(log(5)/(2*m^2*Pi)) then f(a(n)) = n, for n > 0. - Miko Labalan, Oct 07 2024

References

  • Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 603-604.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A035343, A349936 (bisection).
Row 5 of A077042. Column 2 of A201551 (?)

Programs

  • GAP
    List([0..25],n->Sum([0..Int(2*n/5)],k->Binomial(n,k)*Binomial(-n,2*n-5*k))); # Muniru A Asiru, Sep 26 2018
  • Maple
    seq(coeff(series(((1-x^10)/((1-x^5)*(1-x^2)*(1-x)))^n,x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 26 2018
  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,4}]^n],x^(2*n)],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)
a[n_] := a[n] = Sum[n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!), {i, 0, n/2}, {j, 0, n}, {q, n, 2*n}]; Table[a[n], {n, 0, 29}] (* Zagros Lalo, Sep 25 2018 *)
CoefficientList[Series[Sqrt[(-5x+2+2Sqrt[5x^2-6x+1])/(25x^3-10x^2-19x+4)],{x,0,30}],x] (* Harvey P. Dale, Aug 04 2021 *)
  • PARI
    a(n)=if(n<0,0,polcoeff(((1-x^5)/(1-x)+x*O(x^(2*n)))^n,2*n))
  • PARI
    a(n)=if(n<0,0,polcoeff(((1-x^10)/((1-x^5)*(1-x^2)*(1-x))+x*O(x^n))^n,n))
  • PARI
    a(n) = sum(k=0,(2*n)\5,binomial(n,k)*binomial(-n,2*n-5*k)) /* Max Alekseyev */
  • PARI
    a(n) = round((5^n+sum(j=1,2*n-1,(sin(5*Pi*j/2/n)/sin(Pi*j/2/n))^n))/2/n)-2 /* Max Alekseyev */
  • PARI
    a(n) = vecmax(Vec(Pol(vector(5,k,1))^n)); \\ Michel Marcus, Jan 29 2017

Formula

a(n) = Sum_{k=0..floor(2n/5)} binomial(n,k)*binomial(-n, 2n-5k); a(n) = (5^n + Sum_{j=1..2n-1} (sin(5j*Pi/(2n))/sin(j*Pi/(2n)))^n)/(2n) - 2. - Max Alekseyev, Mar 04 2005
D-finite with recurrence: 2*n*(2*n-1)*(3*n-4)*a(n) - (3*n-1)*(19*n^2-38*n+18)*a(n-1) - 5*(n-1)*(3*n-4)*(2*n-1)*a(n-2) + 25*(n-1)*(n-2)*(3*n-1)*a(n-3) = 0. - R. J. Mathar, Feb 21 2010 [Proved using the Almkvist-Zeilberger algorithm in EKHAD. - Doron Zeilberger, Apr 02 2013]
G.f.: sqrt((-5*x+2+2*sqrt(5*x^2-6*x+1))/(25*x^3-10*x^2-19*x+4)). - Mark van Hoeij, May 06 2013
a(n) ~ 5^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013
a(n) = Sum_{i=0..n/2} Sum_{j=0..n} Sum_{q=n..2*n}(f); f=( n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!) ); f=0 for (j - 2*q + 2*n)<0 or (i - 2*j + q)<0 or (j - 2*i)<0. Also see formula in Links section. - Zagros Lalo, Sep 25 2018

A063260 Sextinomial (also called hexanomial) coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Comments

The sequence of step width of this staircase array is [1,5,5,...], hence the degree sequence for the row polynomials is [0,5,10,15,...]=A008587.
The column sequences (without leading zeros) are for k=0..5 those of the lower triangular array A007318 (Pascal) and for k=6..9: A062989, A063262-4. Row sums give A000400 (powers of 6). Central coefficients give A063419; see also A018901.
This can be used to calculate the number of occurrences of a given roll of n six-sided dice, where k is the index: k=0 being the lowest possible roll (i.e., n) and n*6 being the highest roll.

Examples

			The irregular table T(n, k) begins:
n\k 0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15
1:  1
2:  1 1 1  1  1  1
3:  1 2 3  4  5  6  5  4  3  2  1
4:  1 3 6 10 15 21 25 27 27 25 21 15 10  6  3  1
...reformatted - _Wolfdieter Lang_, Oct 31 2015

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

Links

Crossrefs

The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651.
Columns for k=0..9 (with some shifts) are: A000012, A000027, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.
Cf. A000400, A008587, A018901, A063261, A063419.

Programs

  • Maple
    #Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 6-nomials as a table
r := 6:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
  • Mathematica
    Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)
  • PARI
    concat(vector(5,k,Vec(sum(j=0,5,x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Formula

G.f. for row n: (Sum_{j=0..5} x^j)^n.
G.f. for column k: (x^(ceiling(k/5)))*N6(k, x)/(1-x)^(k+1) with the row polynomials from the staircase array A063261(k, m) and with N6(6,x) = 5 - 10*x + 10*x^2 - 5*x^3 + x^4.
T(n, k) = 0 if n=-1 or k<0 or k >= 5*n + 1; T(0, 0)=1; T(n, k) = Sum_{j=0..5} T(n-1, k-j) else.
T(n, k) = Sum_{i = 0..floor(k/6)} (-1)^i*binomial(n,i)*binomial(n+k-1-6*i,n-1) for n >= 0 and 0 <= k <= 5*n. - Peter Bala, Sep 07 2013
T(n, k) = Sum_{i = max(0,ceiling((k-2*n)/3)).. min(n,k/3)} binomial(n,i)*trinomial(n,k-3*i) for n >= 0 and 0 <= k <= 5*n. - Matthew Monaghan, Sep 30 2015

Extensions

More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002

A063265 Septinomial (also called heptanomial) coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 116, 149, 180, 206, 224, 231, 224, 206, 180, 149, 116, 84, 56, 35
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Comments

The sequence of step width of this staircase array is [1,6,6,...], hence the degree sequence for the row polynomials is [0,6,12,18,...]= A008588.
The column sequences (without leading zeros) are for k=0..6 those of the lower triangular array A007318 (Pascal) and for k=7..9: A063267, A063417, A063418. Row sums give A000420 (powers of 7). Central coefficients give A025012.

Examples

			Triangle begins:
  {1};
  {1, 1, 1, 1, 1, 1, 1};
  {1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1};
  ...
N7(k,x)= 1 for k=0..6, N7(7,x)= 6-15*x+20*x^2-15*x^3+6*x^4-x^5 (from A063266).

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

Links

Crossrefs

The q-nomial arrays are for q=2..8: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890.

Programs

  • Maple
    #Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 7-nomials as a table
r := 7:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
  • Mathematica
    Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)

Formula

a(n, k)=0 if n=-1 or k<0 or k >= 6*n; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..6) else.
G.f. for row n: (sum(x^j, j=0..6))^n.
G.f. for column k: (x^(ceiling(k/6)))*N7(k, x)/(1-x)^(k+1) with the row polynomials of the staircase array A063266(k, m).
T(n,k) = Sum_{i = 0..floor(k/7)} (-1)^i*binomial(n,i)*binomial(n+k-1-7*i,n-1) for n >= 0 and 0 <= k <= 6*n. - Peter Bala, Sep 07 2013

A171890 Octonomial coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35
Offset: 0

Views

Author

N. J. A. Sloane, Oct 19 2010

Keywords

Comments

Row lengths are 1,8,15,22,... = 1+7n = A016993(n). Row sums are 1,8,64,... = 8^n = A001018(n). M. F. Hasler, Jun 17 2012

Examples

			Array begins:
[1]
[1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
...

Links

Crossrefs

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287,A035343, A063260, A063265, A171890, A213652, A213651.

Programs

  • Maple
    #Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 8-nomials as a table
r := 8:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
  • Mathematica
    Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)
  • PARI
    concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Formula

Row n has g.f. (1+x+...+x^7)^n.
T(n,k) = sum {i = 0..floor(k/8)} (-1)^i*binomial(n,i)*binomial(n+k-1-8*i,n-1) for n >= 0 and 0 <= k <= 7*n. - Peter Bala, Sep 07 2013

A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

Original entry on oeis.org

1, 5, 5, 7, 5, 17, 7, 19, 5, 25, 17, 19, 7, 31, 19, 25, 5, 25, 25, 35, 17, 61, 19, 71, 7, 35, 31, 41, 19, 71, 25, 77, 5, 25, 25, 35, 25, 85, 35, 95, 17, 85, 61, 71, 19, 91, 71, 77, 7, 35, 35, 49, 31, 107, 41, 121, 19, 95, 71, 85, 25, 113, 77, 103
Offset: 0

Views

Author

N. J. A. Sloane, Sep 25 2014

Keywords

Comments

This is the number of cells that are ON after n generations in a one-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of 5 contiguous cells.
a(n) is also the number of odd entries in row n of A035343. - Leon Rische, Feb 02 2023

Examples

			The first few generations are:
..........X..........
........XXXXX........
......X.X.X.X.X......
....XX..X.X.X..XX.... (f^3)
..X...X...X...X...X..
XXXX.XXX.XXX.XXX.XXXX
...
f^3 mod 2 = x^6 + x^5 + x^2 + 1/x^2 + 1/x^5 + 1/x^6 + 1 has 7 terms, so a(3) = 7.
From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
  1;
  5;
  5, 7;
  5,17, 7,19;
  5,25,17,19, 7,31,19,25;
  5,25,25,35,17,61,19,71, 7,35,31,41,19,71,25,77;
  5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19, ...
(End)
It follows from the Generalized Run Length Transform result mentioned in the comments that in each row the first quarter of the terms (and no more) are equal to 5 times the beginning of the sequence itself. It cannot be said that the rows converge (in any meaningful sense) to five times the sequence. - _N. J. A. Sloane_, Mar 03 2015

Links

Crossrefs

Cf. A071053, A247647, A247648, A253085, A255490.
Partial sums are in A255654.

Programs

  • Python
    import sympy
from functools import reduce
from operator import mul
x = sympy.symbols('x')
f = 1/x**2+1/x+1+x+x**2
A247649_list, g = [1], 1
for n in range(1,1001):
    s = [int(d,2) for d in bin(n)[2:].split('00') if d != '']
    g = (g*f).expand(modulus=2)
    if len(s) == 1:
        A247649_list.append(g.subs(x,1))
    else:
        A247649_list.append(reduce(mul,(A247649_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014

Formula

The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 5*17 = 85. This is a generalization of the Run Length Transform.

A213651 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 282, 348, 415, 480
Offset: 0

Views

Author

M. F. Hasler, Jun 17 2012

Keywords

Comments

The n-th row also yields the number of ways to get a total of n, n+1, ..., 10n, when throwing n 10-sided dice, or summing n integers ranging from 1 to 10.
The row sums equal 10^n = A011557(n).
The row lengths are 1 + 9n = 10n - (n-1) = A017173(n).
T(n,k) is the number of integers in the [0, 10^n-1] range distributed according to the sum k of their digits. - Miquel Cerda, Jun 21 2017
The sum of the squares of the integers of the n-th row gives A174061(n). - Miquel Cerda, Jul 03 2017

Examples

			There are 1, 3, 6, 10, ... ways to score a total of 4, 5, 6, 7, ... when throwing three 10-sided dice.
The table begins as follows:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1,1; (row sum = 10, row length = 10)
(row n=2) 1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1; (sum = 100, length = 19)
(row n=3) 1,3,6,10,15,21,28,36,45,55,63,69,73,75,75,73,...; row sum = 1000;
(row n=4) 1,4,10,20,35,56,84,120,165,220,282,348,415,...; row sum = 10^4;
etc.
Number of integers in (row n=2): k(2)=3, because in the range 0 to 99 there are 3 integers whose digits sum to 2: 2, 11 and 20. - _Miquel Cerda_, Jun 21 2017

Links

Crossrefs

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Programs

  • Maple
    #Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 10-nomials as a table
r := 10:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
  • PARI
    concat(vector(5,k,Vec(sum(j=0,9,x^j)^(k-1))))

Formula

T(n,k) = Sum_{i = 0..floor(k/10)} (-1)^i*binomial(n,i)*binomial(n+k-1-10*i,n-1) for n >= 0 and 0 <= k <= 9*n. - Peter Bala, Sep 07 2013

A187925 Coefficient of x^n in (1+x+x^2+x^3+x^4)^n.

Original entry on oeis.org

1, 1, 3, 10, 35, 121, 426, 1520, 5475, 19855, 72403, 265233, 975338, 3598180, 13311000, 49360405, 183424355, 682870587, 2546441085, 9509714340, 35561166195, 133138728845, 499005557515, 1872137642125, 7030166054250, 26421479140746, 99376657487396
Offset: 0

Views

Author

Emanuele Munarini, Mar 16 2011

Keywords

Links

Crossrefs

Cf. A035343, A005191, A036766.
Column k=4 of A305161.

Programs

  • Magma
    P:=PolynomialRing(Integers()); [Coefficients((1+x+x^2+x^3+x^4)^n)[n+1]: n in [0..26]]; // Vincenzo Librandi, Aug 09 2014
  • Mathematica
    Pentanomial[n_, k_] := If[k == 0, 1, Coefficient[(1 + x + x^2 + x^3 + x^4)^n, x^k]]
Table[Pentanomial[n, n], {n, 0, 12}]
  • Maxima
    pentanomial(n,k):=coeff(expand((1+x+x^2+x^3+x^4)^n),x,k);
makelist(pentanomial(n,n),n,0,12);
  • Maxima
    a(n):=sum(binomial(n,r)*sum((sum(binomial(j,-r+n-m-j)*binomial(m,j),j,0,m))*binomial(r,m),m,0,r),r,0,n); /* Vladimir Kruchinin, Feb 03 2013 */
  • Maxima
    a(n):=sum((-1)^i*binomial(n,i)*binomial(2*n-5*i-1,n-5*i),i,0,n/5); /* Vladimir Kruchinin, Mar 28 2019 */
  • PARI
    a(n) = polcoeff((1+x+x^2+x^3+x^4)^n, n); \\ Michel Marcus, Aug 09 2014

Formula

a(n) = sum(r=0..n, binomial(n,r)*sum(m=0..r, (sum(j=0..m, binomial(j,-r+n-m-j)*binomial(m,j)))*binomial(r,m))). [Vladimir Kruchinin, Feb 03 2013]
G.f.: 1 + x*G'(x)/G(x) where G(x) is the g.f. of A036766. - Paul D. Hanna, Feb 03 2013
Recurrence: 3*n*(3*n-2)*(3*n-1)*(748*n^3 - 4136*n^2 + 7291*n - 4135)*a(n) = 2*(35156*n^6 - 247126*n^5 + 663756*n^4 - 870079*n^3 + 584710*n^2 - 190393*n + 23280)*a(n-1) + 5*(n-1)*(2244*n^5 - 14652*n^4 + 32367*n^3 - 28501*n^2 + 8564*n - 672)*a(n-2) + 100*(n-2)*(n-1)*(374*n^4 - 1881*n^3 + 2848*n^2 - 1475*n + 222)*a(n-3) + 125*(n-3)*(n-2)*(n-1)*(748*n^3 - 1892*n^2 + 1263*n - 232)*a(n-4). - Vaclav Kotesovec, Feb 11 2015
a(n) ~ c * d^n / sqrt(n), where d = 3.83443724902880556376524112660950145464... is the root of the equation -125-50*d-15*d^2-94*d^3+27*d^4 = 0, c = 0.3404440985692437948910444085315314358395... . - Vaclav Kotesovec, Feb 11 2015
a(n) = Sum_{i=0..n/5} (-1)^i*C(n,i)*C(2*n-5*i-1,n-5*i). - Vladimir Kruchinin, Mar 28 2019
From Peter Bala, Mar 31 2020: (Start)
a(p) == 1 (mod p^2) for any prime p > 5 (follows from Kruchinin's formula above). Cf. A002426. More generally, we may have a(p^k) == a(p^(k-1)) (mod p^(2*k)) for k >= 2 and any prime p.
The sequence b(n) := [x^n] ( F(x)/F(-x) )^n = [x^n] ( F(x)^2/F(x^2) )^n, where F(x) = 1 + x + x^2 + x^3 + x^4, may satisfy the stronger congruences b(p) == 2 (mod p^3) for prime p > 5 (checked up to p = 499). (End)
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