A142978 -id:A142978 - cp's OEIS frontend

A217874 Table A142978 (figurate numbers for n-dimensional cross polytope) extended by a top row.

1, 1, 0, 1, 2, 1, 1, 4, 3, 0, 1, 6, 9, 4, 1, 1, 8, 19, 16, 5, 0, 1, 10, 33, 44, 25, 6, 1, 1, 12, 51, 96, 85, 36, 7, 0, 1, 14, 73, 180, 225, 146, 49, 8, 1, 1, 16, 99, 304, 501, 456, 231, 64, 9, 0, 1, 18, 129, 476, 985, 1182, 833, 344, 81, 10, 1

Offset: 0

M. F. Hasler, Oct 13 2012

Looking at table A142978, it seems natural to extend it by a first row, which turns out to be 1,0,1,0,1,0,.... Indeed, these are the values obtained when the polynomial which defines each of the columns, is "extrapolated" to n=0.

			The table may be written as:
  1,_ v---- the even numbers A005843
  1,_ 0,__ v---- A058331(n) = 2*n^2 + 1.
  1,_ 2,__ 1,__ v---- 4*A006527(n) = 4n(n^2 + 2)/3.
  1,_ 4,__ 3,__ 0,__ v---- 2n^2(n^2 + 5)/3 + 1.
  1,_ 6,__ 9,__ 4,__ 1,___ v---- 2n(2n^4 + 20n^2 + 23)/15.
  1,_ 8,_ 19,_ 16,__ 5,___ 0,
  1, 10,_ 33,_ 44,_ 25,___ 6,__ 1,
  1, 12,_ 51,_ 96,_ 85,__ 36,__ 7,__ 0,
  1, 14,_ 73, 180, 225,_ 146,_ 49,__ 8,_ 1,
  1, 16,_ 99, 304, 501,_ 456, 231,_ 64,_ 9, 0,
  1, 18, 129, 476, 985, 1182, 833, 344, 81, 10, 1,...
Sequence A142978 is the table obtained by deleting the uppermost row 1,0,1,0,1,...
(One could also add a 0th column, with all zeros.)

J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7.

A342824 Number of ways n appears as a cross-polytope number (A142978).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2

Offset: 2

Eric M. Schmidt, Mar 22 2021

nonn

Every entry in the first column (of A142978) is 1, so this sequence starts at a(2).
a(n) is always positive, as the first row lists the positive integers.
a(n) >= 3 infinitely often. This happens, in particular, at every even square > 4. (The second row contains the squares, and the second column the positive even numbers.)
For n <= 10000, the only instance of a(n) > 3 is a(1156) = 4. This occurs because 1156 is even, square, and octahedral (third row of A142978).

Cf. A142978.

def a(n) : return len([K for K in [2..n] if n == next(A142978(N, K) for N in (1..) if A142978(N, K) >= n)])

A048739 Expansion of 1/((1 - x)(1 - 2x - x^2)).

Original entry on oeis.org

1, 3, 8, 20, 49, 119, 288, 696, 1681, 4059, 9800, 23660, 57121, 137903, 332928, 803760, 1940449, 4684659, 11309768, 27304196, 65918161, 159140519, 384199200, 927538920, 2239277041, 5406093003, 13051463048, 31509019100, 76069501249

Offset: 0

Barry E. Williams

Partial sums of Pell numbers A000129.
W(n){1,3;2,-1,1} = Sum_{i=1..n} W(i){1,2;2,-1,0}, where W(n){a,b; p,q,r} implies x(n) = p*x(n-1) - q*x(n-2) + r; x(0)=a, x(1)=b.
Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - R. H. Hardin, Mar 16 2002
Binomial transform of A029744. - Paul Barry, Apr 23 2004
Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba, Jun 16 2004
Equals row sums of triangle A153346. - Gary W. Adamson, Dec 24 2008
Equals the sum of the terms of the antidiagonals of A142978. - J. M. Bergot, Nov 13 2012
a(p-2) == 0 mod p where p is an odd prime, see A270342. - Altug Alkan, Mar 15 2016
Also, the lexicographically earliest sequence of positive integers such that for n > 3, {sqrt(2)*a(n)} is located strictly between {sqrt(2)*a(n-1)} and {sqrt(2)*a(n-2)} where {} denotes the fractional part. - Ivan Neretin, May 02 2017
a(n+1) is the number of weak orderings on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n. - J. Devillet, Oct 06 2017

Allombert, Bill, Nicolas Brisebarre, and Alain Lasjaunias. "On a two-valued sequence and related continued fractions in power series fields." The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3, d_{4n+3}.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.
B. Bradie, Extensions and Refinements of some properties of sums involving Pell Numbers, Miss. J. Math. Sci 22 (1) (2010) 37-43
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
Jimmy Devillet, On the single-peakedness property, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1065
Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
Ahmet Öteleş, On the sum of Pell and Jacobsthal numbers by the determinants of Hessenberg matrices, AIP Conference Proceedings 1863, 310003 (2017).
Wipawee Tangjai, A Non-standard Ternary Representation of Integers, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

First row of table A083087.
With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).
Cf. A001333, A048654, A048655, A083044, A083047, A083050, A153346.

a:=n->sum(fibonacci(i,2), i=0..n): seq(a(n), n=1..29); # Zerinvary Lajos, Mar 20 2008

Join[{a=1,b=3},Table[c=2*b+a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1-3x+x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,3,8},30] (* Harvey P. Dale, Jun 13 2011 *)

a(n)=local(w=quadgen(8));-1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n

vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ Altug Alkan, Oct 07 2015

Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ Michel Marcus, May 06 2017

a(n) = 2*a(n-1) + a(n-2) + 1 with n > 1, a(0)=1, a(1)=3.
a(n) = ((2 + (3*sqrt(2))/2)*(1 + sqrt(2))^n - (2 - (3*sqrt(2))/2)*(1 - sqrt(2))^n )/(2*sqrt(2)) - 1/2.
a(0)=1, a(n+1) = ceiling(x*a(n)) for n > 0, where x = 1+sqrt(2). - Paul D. Hanna, Apr 22 2003
a(n) = 3*a(n-1) - a(n-2) - a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, 2k+2)2^k. - Paul Barry, Aug 16 2003
-a(-3-n) = A077921(n). - N. J. A. Sloane, Sep 13 2003
E.g.f.: exp(x)(cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2. - N. J. A. Sloane, Sep 13 2003
a(n) = floor((1+sqrt(2))^(n+2)/4). - Bruno Berselli, Feb 06 2013
a(n) = (((1-sqrt(2))^(n+2) + (1+sqrt(2))^(n+2) - 2) / 4). - Altug Alkan, Mar 16 2016
2*a(n) = A001333(n+2)-1. - R. J. Mathar, Oct 11 2017
a(n) = Sum_{k=0..n} binomial(n+1,k+1)*2^floor(k/2). - Tony Foster III, Oct 12 2017

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A014820 a(n) = (1/3)(n^2 + 2n + 3)*(n+1)^2.

Original entry on oeis.org

1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, 7008, 9633, 12936, 17025, 22016, 28033, 35208, 43681, 53600, 65121, 78408, 93633, 110976, 130625, 152776, 177633, 205408, 236321, 270600, 308481

Offset: 0

N. J. A. Sloane

a(n) is the number of 4 X 4 pandiagonal magic squares with sum 2n. - Sharon Sela (sharonsela(AT)hotmail.com), May 10 2002
Figurate numbers based on the 4-dimensional regular convex polytope called the 16-cell, hexadecachoron, 4-cross polytope or 4-hyperoctahedron with Schlaefli symbol {3,3,4}. a(n)=(n^2*(n^2+2))/3 if the offset were 1. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 18 2009
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 7-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007
Equals binomial transform of [1, 7, 18, 20, 8, 0, 0, 0, ...], where (1, 7, 18, 20, 8) = row 4 of the Chebyshev triangle A081277. Also = row 4 of the array in A142978. - Gary W. Adamson, Jul 19 2008

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Ahmed, J. De Loera and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXiv:math/0201108 [math.CO], 2002.
Maya Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral cones of magic cubes and squares, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Eric Weisstein's World of Mathematics, 16-Cell
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000332, A000583, A005900, A069038, A069039, A070212, A081277, A092181, A092182, A092183, A099175, A099193, A099195, A099196, A099197, A142978.

Cf. A061927, A002415, A000290, A039623.

List([0..40], n -> (n+1)^2*((n+1)^2 +2)/3); # G. C. Greubel, Feb 10 2019

[(1/3)*(n^2+2*n+3)*(n+1)^2: n in [0..40]]; // Vincenzo Librandi, May 22 2011

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n),r=0..d-1); end; [seq(be(4,n),n=0..100)];

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 33, 96, 225}, 31] (* Jean-François Alcover, Jan 17 2018 *)

a(n)=(n+1)^2*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Apr 17 2012

a <- c(1, 8, 33, 96,225)
for(n in (length(a)+1):30) a[n] <- 5*a[n-1]-10*a[n-2]+10*a[n-3]-5*a[n-4]+a[n-5]
a # Yosu Yurramendi, Sep 03 2013

[((n+1)^2+2)*(n+1)^2/3 for n in range(40)] # G. C. Greubel, Feb 10 2019

Or, a(n-1) = n^2*(n^2+2)/3. - Corrected by R. J. Mathar, Jul 18 2009
From Vladeta Jovovic, Apr 03 2002: (Start)
G.f.: (1+x)^3/(1-x)^5.
Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
a(n-1) = C(n+3,4) + 3 C(n+2,4) + 3 C(n+1,4) + C(n,4).
Sum_{n>=0} 1/((1/3*(n^2 + 2*n + 3))*(n+1)^2) = (1/4)*Pi^2 - 3*sqrt(2)*Pi*coth(Pi*sqrt(2))*(1/8) + 3/8 = 1.1758589... - Stephen Crowley, Jul 14 2009
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with n > 4, a(0)=1, a(1)=8, a(2)=33, a(3)=96, a(4)=225. - Yosu Yurramendi, Sep 03 2013
From Bruce J. Nicholson, Jan 23 2019: (Start)
Sum_{i=0..n} a(i) = A061927(n+1).
a(n) = 4*A002415(n+1) + A000290(n+1) = A039623(n+1) + A002415(n+1). (End)
E.g.f.: (3 + 21*x + 27*x^2 + 10*x^3 + x^4)*exp(x)/3. - G. C. Greubel, Feb 10 2019
Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 17/3 - 8*log(2) = 1/(8 + 2/(8 + 6/(8 + ... + n*(n-1)/(8 + ...)))). See A142983. - Peter Bala, Mar 06 2024

Formula index corrected by R. J. Mathar, Jul 18 2009

A047781 a(n) = Sum_{k=0..n-1} binomial(n-1,k)*binomial(n+k,k). Also a(n) = T(n,n), array T as in A049600.

Original entry on oeis.org

0, 1, 4, 19, 96, 501, 2668, 14407, 78592, 432073, 2390004, 13286043, 74160672, 415382397, 2333445468, 13141557519, 74174404608, 419472490257, 2376287945572, 13482186743203, 76598310928096, 435730007006341, 2481447593848524, 14146164790774359

Offset: 0

N. J. A. Sloane

nonn

Also main diagonal of array: m(i,1)=1, m(1,j)=j, m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 3 4 ... / 1 4 9 16 ... / 1 6 19 44 ... / 1 8 33 96 ... /. - Benoit Cloitre, Aug 05 2002
This array is now listed as A142978, where some conjectural congruences for the present sequence are given. - Peter Bala, Nov 13 2008
Convolution of central Delannoy numbers A001850 and little Schroeder numbers A001003. Hankel transform is 2^C(n+1,2)*A007052(n). - Paul Barry, Oct 07 2009
Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0 <= r <= n and the other terms recursively with T(r,c) = T(r-1,c-1) + 2*T(r,c-1). The sum of the last terms in the rows is Sum_{r=0..n} T(r,r) = a(n+1). Example: For n=4 the triangle has the rows 1; 4 9; 6 16 41; 4 14 44 129; 1 6 26 96 321 having sum of last terms 1 + 9 + 41 + 129 + 321 = 501 = a(5). - J. M. Bergot, Feb 15 2013
a(n) = A049600(2*n,n), when A049600 is seen as a triangle read by rows. - Reinhard Zumkeller, Apr 15 2014
a(n-1) for n > 1 is the number of assembly trees with the connected gluing rule for cycle graphs with n vertices. - Nick Mayers, Aug 16 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
A. Bacher, Directed and multi-directed animals on the square lattice with next nearest neighbor edges, arXiv preprint arXiv:1301.1365 [math.CO], 2013. See D(t). - From _N. J. A. Sloane_, Feb 14 2013
C. Banderier and P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 2.
M. Bona and A. Vince, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
A. Dougherty, N. Mayers, and R. Short, How to Build a Graph in n Days: Some Variants on Graph Assembly, arXiv preprint arXiv:1807.08079 [math.CO], 2018.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Milan Janjic, Two Enumerative Functions
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Cf. A002003. Column 1 of A296129.

a047781 n = a049600 (2 * n) n  -- Reinhard Zumkeller, Apr 15 2014

[n eq 0 select 0 else &+[Binomial(n-1, k)*Binomial(n+k, k): k in [0..n-1]]: n in [0..22]];  // Bruno Berselli, May 19 2011

a := proc(n) local k; add(binomial(n-1,k)*binomial(n+k,k),k=0..n-1); end;

Table[SeriesCoefficient[x*((1+x)-Sqrt[1-6*x+x^2])/(4*x*Sqrt[1-6*x+x^2]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] := Hypergeometric2F1[1-n, n+1, 1, -1]; a[0] = 0; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 26 2013 *)
a[n_] := Sum[ Binomial[n - 1, k] Binomial[n + k, k], {k, 0, n - 1}]; Array[a, 25] (* Robert G. Wilson v, Aug 08 2018 *)

makelist(if n=0 then 0 else sum(binomial(n-1, k)*binomial(n+k, k), k, 0, n-1), n, 0, 22); /* Bruno Berselli, May 19 2011 */

A047781(n)=polcoeff((1+x)/sqrt(1+(O(x^n)-6)*x+x^2),n)\4  \\ M. F. Hasler, Oct 09 2012

from sympy import binomial
def a(n):
    return sum(binomial(n - 1, k) * binomial(n + k, k) for k in range(n))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017

from math import comb
def A047781(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

D-finite with recurrence n*(2*n-3)*a(n) - (12*n^2-24*n+8)*a(n-1) + (2*n-1)*(n-2)*a(n-2) = 0. - Vladeta Jovovic, Aug 29 2004
a(n+1) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)*2^k. - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..n} T(n, k), array T as in A008288.
If shifted one place left, the third binomial transform of A098660. - Paul Barry, Sep 20 2004
G.f.: ((1+x)/sqrt(1-6x+x^2)-1)/4. - Paul Barry, Sep 20 2004, simplified by M. F. Hasler, Oct 09 2012
E.g.f. for sequence shifted left: Sum_{n>=0} a(n+1)*x^n/n! = exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+BesselI(1, 2*sqrt(2)*x)/sqrt(2)). - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*2^(n-k-1); a(n+1) = 2^n*Hypergeometric2F1(-n,-n-1;1;1/2). - Paul Barry, Feb 08 2011
a(n) ~ 2^(1/4)*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = A241023(n) / 4. - Reinhard Zumkeller, Apr 15 2014
a(n) = Hyper2F1([-n, n], [1], -1)/2 for n > 0. - Peter Luschny, Aug 02 2014
n^2*a(n) = Sum_{k=0..n-1} (2*k^2+2*k+1)*binomial(n-1,k)*binomial(n+k,k). By the Zeilberger algorithm, both sides of the equality satisfy the same recurrence. - Zhi-Wei Sun, Aug 30 2014
a(n) = [x^n] (1/2) * ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018

A081277 Square array of unsigned coefficients of Chebyshev polynomials of the first kind.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 8, 4, 1, 7, 18, 20, 8, 1, 9, 32, 56, 48, 16, 1, 11, 50, 120, 160, 112, 32, 1, 13, 72, 220, 400, 432, 256, 64, 1, 15, 98, 364, 840, 1232, 1120, 576, 128, 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 1, 19, 162, 816, 2688, 6048, 9408, 9984, 6912

Offset: 0

Paul Barry, Mar 16 2003

Rows include A011782, A001792, A001793, A001794, A006974.
Formatted as a triangular array, this is [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] (see construction in A084938 ). - Philippe Deléham, Aug 09 2005
Antidiagonal sums are in A025192. - Philippe Deléham, Dec 04 2006
Binomial transform of n-th row of the triangle (followed by zeros) = n-th row of the A142978 array and n-th column of triangle A104698. - Gary W. Adamson, Jul 17 2008
When formatted as a triangle, A038763=fusion of polynomial sequences (x+1)^n and (x+1)^n; see A193722 for the definition of fusion of two polynomial sequences or triangular arrays.  Row n of A038763, as a triangle, consists of coefficients of the product (x+1)*(x+2)^n. - Clark Kimberling, Aug 04 2011

			Rows begin
  1, 1,  2,   4,   8, ...
  1, 3,  8,  20,  48, ...
  1, 5, 18,  56, 160, ...
  1, 7, 32, 120, 400, ...
  1, 9, 50, 220, 840, ...
  ...
As a triangle:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  8,  4;
  1,  7, 18, 20,  8;

Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.

Cf. A079628.
Cf. A142978, A104698.
Cf. A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
Cf. A053120 (antidiagonals give signed version) and A124182 (skewed version). - Mathias Zechmeister, Jul 26 2022

(* Program generates triangle A081277 as the self-fusion of Pascal's triangle *)
z = 8; a = 1; b = 1; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A081277 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* abs val of A118800 *)
Factor[w[6, x]]
(* Clark Kimberling, Aug 04 2011 *)

T(n, k) = (n+2k)*binomial(n+k-1, k-1)*2^(n-1)/k, k > 0.
T(n, 0) defined by g.f. (1-x)/(1-2x). Other rows are defined by (1-x)/(1-2x)^n.
T(n, 0) = 0 if n < 0, T(0, k) = 0 if k < 0, T(0, 0) = T(1, 0) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k); for example, 160 = 48 + 2*56 for n = 4 and k = 2. -Philippe Deléham, Aug 12 2005
G.f. of the triangular interpretation: (-1+x*y)/(-1+2*x*y+x). - R. J. Mathar, Aug 11 2015

A142983 a(1) = 1, a(2) = 2, a(n+2) = 2a(n+1) + (n + 1)(n + 2)*a(n).

Original entry on oeis.org

1, 2, 10, 44, 288, 1896, 15888, 137952, 1419840, 15255360, 186693120, 2387093760, 33898314240, 502247692800, 8123141376000, 136785729024000, 2483065912320000, 46822564905984000, 942853671825408000, 19678282007924736000, 435355106182520832000

Offset: 1

Peter Bala, Jul 17 2008

This is the case m = 1 of the general recurrence a(1) = 1, a(2) = 2*m, a(n+2) = 2*m*a(n+1) + (n + 1)*(n + 2)*a(n) (we suppress the dependence of a(n) on m), which arises when accelerating the convergence of the series 1/2 + 1/2*Sum_{k > 1} (-1)^(k+1)/(k*(k + 1)) = log(2). For other cases see A142984 (m = 2), A142985 (m = 3), A142986 (m = 4) and A142987 (m = 5).
The solution to the general recurrence may be expressed as a sum: a(n) = n!*p_m(n+1)*Sum_{k = 1..n} (-1)^(k+1)/(p_m(k)*p_m(k+1)), where p_m(x) = Sum_{k = 1..m} 2^(k-1)*C(m-1,k-1)*C(x,k) is the polynomial that gives the regular polytope numbers for the m-dimensional cross polytope as defined by [Kim] (see A142978). The first few values are p_1(x) = x, p_2(x) = x^2, p_3(x) = (2*x^3 + x)/3 and p_4(x) = (x^4 + 2*x^2)/3.
The polynomial p_m(x) is the unique polynomial solution of the difference equation x*(f(x+1) - f(x-1)) = 2*m*f(x), normalized so that f(1) = 1.
The o.g.f. for the p_m(x) is 1/2*((1 + t)/(1 - t))^x = 1/2 + x*t + x^2*t^2 + (2*x^3 + x)/3*t^3 + .... Thus p_m(x) is, apart from a constant factor, the Meixner polynomial of the first kind M_m(x;b,c) at b = 0, c = -1, also known as a Mittag-Leffler polynomial.
The general recurrence in the first paragraph above has a second solution b(n) = n!*p_m(n+1) with b(1) = 2*m, b(2) = m^2 + 2. Hence the behavior of a(n) for large n is given by Limit_{n-> oo} a(n)/b(n) = Sum_{k >= 1} (-1)^(k+1)/(p_m(k)*p_m(k+1)) = 1/((2*m) + 1*2/((2*m) + 2*3/((2*m) + 3*4/((2*m) + ... + n*(n + 1)/((2*m) + ...))))) = 1 + (-1)^(m+1) * (2*m)*(log(2) - (1 - 1/2 + 1/3 - ... + (-1)^(m+1)/m)), where the final equality follows by a result of Ramanujan (see [Berndt, Chapter 12, Entry 32(i)]).
See A142979, A142988 and A142992 for similar results. For corresponding results for Napier's constant e, the constant zeta(2) and Apery's constant zeta(3) refer to A000522, A142995 and A143003, respectively.

Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Eric Weisstein's World of Mathematics, Meixner polynomial of the first kind.
Eric Weisstein's World of Mathematics, Mittag-Leffler polynomial.

Cf. A000522, A142979, A142984, A142985, A142986, A142987, A142988, A142992.

Cf. A002378.

a142983 n = a142983_list !! (n-1)
a142983_list = 1 : 2 : zipWith (+)
                       (map (* 2) $ tail a142983_list)
                       (zipWith (*) (drop 2 a002378_list) a142983_list)
-- Reinhard Zumkeller, Jul 17 2015

a := n -> (n+1)!*sum ((-1)^(k+1)/(k*(k+1)), k = 1..n): seq(a(n), n = 1..20);

Rest[CoefficientList[Series[(-x+2*Log[x+1])/(x-1)^2,{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Oct 21 2012 *)

a(n) = n!*p(n+1)*Sum_{k = 1..n} (-1)^(k+1)/(p(k)*p(k+1)), where p(n) = n.
Recurrence: a(1) = 1, a(2) = 2, a(n+2) = 2*a(n+1) + (n + 1)*(n + 2)*a(n).
The sequence b(n) := n!*p(n+1) satisfies the same recurrence with b(1) = 2 and b(2) = 6.
Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(2 + 1*2/(2 + 2*3/(2 + 3*4/(2 + ... + (n - 1)*n/2)))), for n >= 2.
The behavior of a(n) for large n is given by Limit_{n -> oo} a(n)/b(n) = 1/(2 + 1*2/(2 + 2*3/(2 + 3*4/(2 + ... + n*(n+1)/(2 + ...))))) = Sum_{k >= 1} (-1)^(k+1)/(k*(k + 1)) = 2*log(2) - 1.
E.g.f.: (2*log(x+1)-x)/(x-1)^2. - Vaclav Kotesovec, Oct 21 2012

A069038 Expansion of g.f. x*(1+x)^4/(1-x)^6.

Original entry on oeis.org

0, 1, 10, 51, 180, 501, 1182, 2471, 4712, 8361, 14002, 22363, 34332, 50973, 73542, 103503, 142544, 192593, 255834, 334723, 432004, 550725, 694254, 866295, 1070904, 1312505, 1595906, 1926315, 2309356, 2751085, 3258006, 3837087, 4495776, 5242017, 6084266

Offset: 0

Vladeta Jovovic, Apr 03 2002

Hyun Kwang Kim asserts that every nonnegative integer can be represented by the sum of no more than 14 of these numbers. - Jonathan Vos Post, Nov 16 2004
If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Oct 28 2007
Starting with 1 = binomial transform of [1, 9, 32, 56, 48, 16, 0, 0, 0, ...], where (1, 9, 32, 56, 48, 16) = row 5 of the Chebyshev triangle A081277. Also = row 5 of the array in A142978. - Gary W. Adamson, Jul 19 2008
Starting with the term 1 this is the self-convolution of A001844(n). - Anton Zakharov, Sep 02 2016

H. S. M. Coxeter, Regular Polytopes, New York: Dover Publications, 1973.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000332, A014820, A005900.
Cf. A142978, A081277.
Cf. A001844.

[n*(2*n^4 + 10*n^2 + 3)/15: n in [0..40]]; // Vincenzo Librandi, May 22 2011

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(5,n),n=0..100)];

CoefficientList[Series[x (1 + x)^4/(1 - x)^6, {x, 0, 32}], x] (* Michael De Vlieger, Sep 02 2016 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,10,51,180,501},40] (* Harvey P. Dale, Jun 19 2021 *)

concat(0, Vec(x*(1+x)^4/(1-x)^6 + O(x^99))) \\ Altug Alkan, Sep 02 2016

{a(n) = n * (2*n^4 + 10*n^2 + 3) / 15}; /* Michael Somos, Jun 17 2018 */

Recurrence: a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = n*(2*n^4 + 10*n^2 + 3)/15. - Jonathan Vos Post, Nov 16 2004
a(n) = C(n+4,5) + 4*C(n+3,5) + 6*C(n+2,5) + 4*C(n+1,5) + C(n,5).
Sum_{n>=1} 1/((1/15)*n*(2*n^4 + 10*n^2 + 3)*n!) = hypergeom([1, 1, 1+i*sqrt(10-2*sqrt(19))*(1/2), 1-i*sqrt(10-2*sqrt(19))*(1/2), 1+i*sqrt(10+2*sqrt(19))*(1/2), 1-i*sqrt(10+2*sqrt(19))*(1/2)], [2, 2, 2+i*sqrt(10-2*sqrt(19))*(1/2), 2-i*sqrt(10-2*sqrt(19))*(1/2), 2+i*sqrt(10+2*sqrt(19))*(1/2), 2-i*sqrt(10+2*sqrt(19))*(1/2)], 1) = 1.05351734968093116819345664995829700099916... - Stephen Crowley, Jul 14 2009
a(n) = a(n-1) + A014820(n-1) + A014820(n-2). - Bruce J. Nicholson, Apr 18 2018
a(n) = 10*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
Euler transform of length 2 sequence [10, -4]. - Michael Somos, Jun 19 2018
Sum_{k >= 1} (-1)^k/(a(k)*a(k+1)) = 10*log(2) - 41/6 = 1/(10 + 2/(10 + 6/(10 + ... + n*(n-1)/(10 + ...)))). See A142983. Cf. A005900 and A014820. - Peter Bala, Mar 08 2024
E.g.f.: exp(x)*x*(15 + 60*x + 60*x^2 + 20*x^3 + 2*x^4)/15. - Stefano Spezia, Mar 10 2024

A069039 Expansion of g.f. x*(1+x)^5/(1-x)^7.

Original entry on oeis.org

0, 1, 12, 73, 304, 985, 2668, 6321, 13504, 26577, 48940, 85305, 142000, 227305, 351820, 528865, 774912, 1110049, 1558476, 2149033, 2915760, 3898489, 5143468, 6704017, 8641216, 11024625, 13933036, 17455257, 21690928, 26751369, 32760460, 39855553, 48188416, 57926209

Offset: 0

Vladeta Jovovic, Apr 03 2002

Figurate numbers based on the 6-dimensional regular convex polytope called the 6-dimensional cross-polytope, or 6-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 4}. It is the dual of the 6-dimensional hypercube. Kim asserts that every nonnegative integer can be represented by the sum of no more than 19 of these 6-crosspolytope numbers. - Jonathan Vos Post, Nov 16 2004

Starting with 1 = binomial transform of [1, 11, 50, 120, 160, 112, 32, 0, 0, 0, ...] where (1, 11, 50, 120, 160, 112, 32) = row 6 of the Chebyshev triangle A081277. Also = row 6 of the array in A142978. - Gary W. Adamson, Jul 19 2008

H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 240.
Jonathan Vos Post, "4-Dimensional Jonathan numbers: polytope numbers and Centered polytope numbers of Higher Than 3 Dimensions", Draft 1.5 of 9 a.m., 12 March 2004, circulated by e-mail.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of polytope numbers, Sorted, Through 1,000,000.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Similar sequence: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A099193 (m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).
Cf. A000332.
Cf. A081277, A142978.

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(6,n),n=0..100)];

a[n_] := n^2*(2*n^4 + 20*n^2 + 23)/45; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 29 2014 *)
CoefficientList[Series[x (1+x)^5/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,1,12,73,304,985,2668},40] (* Harvey P. Dale, Aug 05 2018 *)

x='x+O('x^100); concat(0, Vec(x*(1+x)^5/(1-x)^7)) \\ Altug Alkan, Dec 14 2015

Recurrence: a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
a(n) = (n^2)*(2*n^4 + 20*n^2 + 23 )/45. - Jonathan Vos Post, Nov 16 2004
From Stephen Crowley, Jul 14 2009: (Start)
Sum_{n >= 1} 1/a(n) = -5*(Sum(_alpha*(77*_alpha^2+655)*Psi(1-_alpha), _alpha = RootOf(2*_Z^4+20*_Z^2+23)))*(1/3174)+15*Pi^2*(1/46)=1.10203455013915915542552577192042916250524...
Sum_{n>=1} 1/(a(n)*n!) = hypergeom([1, 1, 1, 1-a, 1+b, 1-b, 1+a], [2, 2, 2, 2+b, 2-b, 2+a, 2-a], 1) = 1.04409584723862654376639417281585634150689... where a = (i/2)*sqrt(20+6*sqrt(6)), b = (i/2)*sqrt(20-6*sqrt(6)), and i = sqrt(-1). (End)
a(n) = 12*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
E.g.f.: exp(x)*x*(45 + 225*x + 300*x^2 + 150*x^3 + 30*x^4 + 2*x^5)/45. - Stefano Spezia, Mar 10 2024

A099193 a(n) = n(4n^6 + 70n^4 + 196n^2 + 45)/315.

Original entry on oeis.org

0, 1, 14, 99, 476, 1765, 5418, 14407, 34232, 74313, 149830, 284075, 511380, 880685, 1459810, 2340495, 3644272, 5529233, 8197758, 11905267, 16970060, 23784309, 32826266, 44673751, 60018984, 79684825, 104642486, 136030779, 175176964, 223619261, 283131090, 355747103

Offset: 0

Jonathan Vos Post, Nov 16 2004

Kim asserts that every nonnegative integer can be represented by the sum of no more than 21 of these numbers.

Starting with 1 = binomial transform of [1, 13, 72, 220, 400, 432, 256, 0, 0, 0, ...], where (1, 13, 72, 220, 400, 432, 256) = row 7 of the Chebyshev triangle A081277. Also = row 7 of the array in A142978. - Gary W. Adamson, Jul 19 2008

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099195 (m=8), A099196 (m=9), A099197 (m=10).
Cf. A000332.
Cf. A142978, A081277.

Table[SeriesCoefficient[x (1 + x)^6/(1 - x)^8, {x, 0, n}], {n, 0, 31}] (* Michael De Vlieger, Dec 14 2015 *)

concat(0, Vec(x*(1+x)^6/(1-x)^8 + O(x^40))) \\ Michel Marcus, Dec 14 2015

a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.
G.f.: x*(1+x)^6/(1-x)^8. - R. J. Mathar, Jul 18 2009
a(n) = 14*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

More terms from Michel Marcus, Dec 14 2015

cp's OEIS Frontend

A217874 Table A142978 (figurate numbers for n-dimensional cross polytope) extended by a top row.

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A342824 Number of ways n appears as a cross-polytope number (A142978).

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A048739 Expansion of 1/((1 - x)*(1 - 2*x - x^2)).

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A014820 a(n) = (1/3)*(n^2 + 2*n + 3)*(n+1)^2.

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A047781 a(n) = Sum_{k=0..n-1} binomial(n-1,k)*binomial(n+k,k). Also a(n) = T(n,n), array T as in A049600.

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A081277 Square array of unsigned coefficients of Chebyshev polynomials of the first kind.

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A142983 a(1) = 1, a(2) = 2, a(n+2) = 2*a(n+1) + (n + 1)*(n + 2)*a(n).

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A048739 Expansion of 1/((1 - x)(1 - 2x - x^2)).

A014820 a(n) = (1/3)(n^2 + 2n + 3)*(n+1)^2.

A142983 a(1) = 1, a(2) = 2, a(n+2) = 2a(n+1) + (n + 1)(n + 2)*a(n).

A099193 a(n) = n(4n^6 + 70n^4 + 196n^2 + 45)/315.