A110555 -id:A110555 - cp's OEIS frontend

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A001287 a(n) = binomial coefficient C(n,10).

Original entry on oeis.org

1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, 44352165, 64512240, 92561040, 131128140, 183579396, 254186856, 348330136, 472733756, 635745396

Offset: 10

N. J. A. Sloane

Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].
Product of 10 consecutive numbers divided by 10!. - Artur Jasinski, Dec 02 2007
In this sequence only 11 is prime. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=10) of 2 objects: u,v, with repetition allowed, containing exactly 10 u's. Example: a(1)=11 because we have uuuuuuuuuuv, uuuuuuuuuvu, uuuuuuuuvuu, uuuuuuuvuuu, uuuuuuvuuuu, uuuuuvuuuuu, uuuuvuuuuuu, uuuvuuuuuuu, uuvuuuuuuuu, uvuuuuuuuuu and vuuuuuuuuuu. - Zerinvary Lajos, Aug 03 2008
a(9+k) is the number of times that each digit appears repeated inside a list made with all the possible base 10 numbers of k digits such that their digits are read in ascending order from left to right. - R. J. Cano Jul 20 2014
a(n) = fallfac(n,10)/10! = binomial(n, 10) is also the number of independent components of an antisymmetric tensor of rank 10 and dimension n >= 10 (for n=1..9 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
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).

T. D. Noe, Table of n, a(n) for n = 10..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 260.
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A110555, A001787, A242091.

[Binomial(n,10): n in [10..40]]; // Vincenzo Librandi, Sep 11 2015

seq(binomial(n,10),n=10..31); # Zerinvary Lajos, Aug 06 2008

Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)/10!, {n, 1, 100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 10], {n, 10, 20}] (* Zerinvary Lajos, Jan 31 2010 *)

a(n)=binomial(n,10) \\ Charles R Greathouse IV, Sep 24 2015

A001287_list, m = [], [1]*11
for _ in range(10**2):
    A001287_list.append(m[-1])
    for i in range(10):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

a(n) = A110555(n+1,10). - Reinhard Zumkeller, Jul 27 2005
a(n+9) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)/10!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^10/(1-x)^11. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
Sum_{k>=10} 1/a(k) = 10/9. - Tom Edgar, Sep 10 2015
Sum_{n>=10} (-1)^n/a(n) = A001787(10)*log(2) - A242091(10)/9! = 5120*log(2) - 447047/126 = 0.9215009748... - Amiram Eldar, Dec 10 2020

Formulas valid for different offsets rewritten by R. J. Mathar, Jul 07 2009

Extended by Ray Chandler, Oct 25 2011

A071919 Number of monotone nondecreasing functions [n]->[m] for n >= 0, m >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 0

Offset: 0

Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002

Sometimes called a Riordan array.
Number of different partial sums of 1 + [2,3] + [3,4] + [4,5] + ... - Jon Perry, Jan 01 2004
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 05 2005
T(n,k)=abs(A110555(n,k)), A110555(n,k)=T(n,k)*(-1)^k. - Reinhard Zumkeller, Jul 27 2005
(1,0)-Pascal triangle. - Philippe Deléham, Nov 21 2006
A129186*A007318 as infinite lower triangular matrices. - Philippe Deléham, Mar 07 2009
Let n>=0 index the rows and m>=0 index the columns of this rectangular array. R(n,m) is "m multichoose n", the number of multisets of length n on m symbols. R(n,m) = Sum_{i=0..n} R(i,m-1). The summation conditions on the number of members in a size n multiset that are not the element m (an arbitrary element in the set of m symbols). R(n,m) = Sum_{i=1..m} R(n-1,i). The summation conditions on the largest element in a size n multiset on {1,2,...,m}. - Geoffrey Critzer, Jun 03 2009
Sum_{k=0..n} T(n,k)*B(k) = B(n), n>=0, with the Bell numbers B(n):=A000110(n) (eigensequence). See, e.g., the W. Lang link, Corollary 4. - Wolfdieter Lang, Jun 23 2010
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			   1,    1,    1,    1,    1,    1,    1,    1,    1, ...
   0,    1,    2,    3,    4,    5,    6,    7,    8, ...
   0,    1,    3,    6,   10,   15,   21,   28,   36, ...
   0,    1,    4,   10,   20,   35,   56,   84,  120, ...
   0,    1,    5,   15,   35,   70,  126,  210,  330, ...
   0,    1,    6,   21,   56,  126,  252,  462,  792, ...
   0,    1,    7,   28,   84,  210,  462,  924, 1716, ...
   0,    1,    8,   36,  120,  330,  792, 1716, 3432, ...
   0,    1,    9,   45,  165,  495, 1287, 3003, 6435, ...

G. C. Greubel, Table of n, a(n) for the first 101 antidiagonals, flattened
D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.
Wolfdieter Lang, Simple proofs of some facts related to the Bell sequence and triangles A007318 (Pascal) and A071919 (enlarged Pascal). [From _Wolfdieter Lang_, Jun 23 2010]

Cf. A000110, A007318, A228196, A228576.
Columns are A000007, A000012, A000027, A000217, A000292, A000332, A000389, ...
Main diagonal gives A088218.

A:= (n, m)-> binomial(n+m-1, n):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Jan 13 2017

Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid (* Geoffrey Critzer, Jun 03 2009 *)
a[n_, m_] := Binomial[m - 1 + n, n]; Table[Table[a[n, m - n], {n, 0, m}], {m, 0, 10}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

{ n=20; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+i; v[i][j+k]=v[i-1][j]+i+1)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

{a(n) = my(m); if( n<1, n==0, m = (sqrtint(8*n+1) - 1)\2; binomial(m-1, n - m*(m+1)/2))}; /* Michael Somos, Aug 20 2006 */

Limit_{k->infinity} A071919^k = (A000110,0,0,0,0,...) with the Bell numbers in the first column. For a proof see, e.g., the W. Lang link, proposition 12.
A(n,k) = binomial(n+k-1,n). - Reinhard Zumkeller, Jul 27 2005
G.f.: 1 + x + x^3(1+x) + x^6(1+x)^2 + x^10(1+x)^3 + ... . - Michael Somos, Aug 20 2006
G.f. of the triangular interpretation: (-1+x*y)/(-1+x*y+x). - R. J. Mathar, Aug 11 2015

A001288 a(n) = binomial(n,11).

Original entry on oeis.org

1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, 84672315, 129024480, 193536720, 286097760, 417225900, 600805296, 854992152, 1203322288, 1676056044

Offset: 11

N. J. A. Sloane

Product of 11 consecutive numbers divided by 11!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=11) of 2 objects: u,v, with repetition allowed, containing exactly (11) u's. Example: n=11, a(0)=1 because we have uuuuuuuuuuu n=12, a(1)=12 because we have uuuuuuuuuuuv, uuuuuuuuuuvu, uuuuuuuuuvuu, uuuuuuuuvuuu, uuuuuuuvuuuu, uuuuuuvuuuuu, uuuuuvuuuuuu, uuuuvuuuuuuu, uuuvuuuuuuuu, uuvuuuuuuuuu uvuuuuuuuuuu, vuuuuuuuuuuu. - Zerinvary Lajos, Aug 06 2008
Does not satisfy Benford's law (because n^11 does not, see Ross, 2012). - N. J. A. Sloane, Feb 09 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
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).

T. D. Noe, Table of n, a(n) for n=11..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. S. Chinchon, Mixing Benford, GoogleVis And On-Line Encyclopedia of Integer Sequences, 2014. Note: as of Feb 09 2017, the results in this page appear to be incorrect - N. J. A. Sloane, Feb 09 2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 261
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Index entries for sequences related to Benford's law
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A110555, A001787, A242091.

seq(binomial(n,11),n=0..30); # Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!,{n,1,100}] (* Artur Jasinski, Dec 02 2007 *)
Binomial[Range[11,50],11] (* Harvey P. Dale, Oct 02 2012 *)

for(n=11, 50, print1(binomial(n,11), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,11). - Reinhard Zumkeller, Jul 27 2005
a(n+10) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^11/(1-x)^12. a(n) = binomial(n,11). - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=11} 1/a(n) = 11/10.
Sum_{n>=11} (-1)^(n+1)/a(n) = A001787(11)*log(2) - A242091(11)/10! = 11264*log(2) - 491821/63 = 0.9273021446... (End)

Some formulas for other offsets corrected by R. J. Mathar, Jul 07 2009

A010965 a(n) = binomial(n,12).

Original entry on oeis.org

1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, 225792840, 354817320, 548354040, 834451800, 1251677700, 1852482996, 2707475148, 3910797436, 5586853480

Offset: 12

N. J. A. Sloane

nonn

Coordination sequence for 12-dimensional cyclotomic lattice Z[zeta_13].

In this sequence only 13 is prime. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 12..1000
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A000581, A010966, A010967, A001787, A242091.

[Binomial(n, 12): n in [12..100]]; // Vincenzo Librandi, Apr 22 2011

seq(binomial(n,12),n=12..36); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,12],{n,12,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=12, 50, print1(binomial(n,12), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A110555(n+1,12). - Reinhard Zumkeller, Jul 27 2005
a(n+11) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)/12!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
G.f.: x^12/(1-x)^13. - Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=12} 1/a(n) = 12/11.
Sum_{n>=12} (-1)^n/a(n) = A001787(12)*log(2) - A242091(12)/11! = 24576*log(2) - 3934820/231 = 0.9322955884... (End)

Some formulas referring to other offsets corrected by R. J. Mathar, Jul 07 2009

A010966 a(n) = binomial(n,13).

Original entry on oeis.org

1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400600, 20058300, 37442160, 67863915, 119759850, 206253075, 347373600, 573166440, 927983760, 1476337800, 2310789600, 3562467300, 5414950296, 8122425444, 12033222880

Offset: 13

N. J. A. Sloane

nonn

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 13..1000
Milan Janjic, Two Enumerative Functions University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,13): n in [13..50]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,13),n=13..36); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,13],{n,13,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=13, 50, print1(binomial(n,13), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,13). - Reinhard Zumkeller, Jul 27 2005
a(n+12) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)/13!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^13/(1-x)^14. - Zerinvary Lajos, Aug 06 2008
a(n) = n/(n-13) * a(n-1), n > 13. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=13} 1/a(n) = 13/12.
Sum_{n>=13} (-1)^(n+1)/a(n) = A001787(13)*log(2) - A242091(13)/12! = 53248*log(2) - 102308323/2772 = 0.9366404415... (End)

Some formulas for different offsets rewritten by R. J. Mathar, Jul 07 2009

A110556 a(n) = binomial(2n-1, n)(-1)^n.

Original entry on oeis.org

1, -1, 3, -10, 35, -126, 462, -1716, 6435, -24310, 92378, -352716, 1352078, -5200300, 20058300, -77558760, 300540195, -1166803110, 4537567650, -17672631900, 68923264410, -269128937220, 1052049481860, -4116715363800, 16123801841550

Offset: 0

Reinhard Zumkeller, Jul 27 2005

			1 - x + 3*x^2 - 10*x^3 + 35*x^4 - 126*x^5 + 462*x^6 - 1716*x^7 + 6435*x^8 - ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Another version of A001700, which is the main entry.

Cf. A000027, A000108, A000984, A007318, A033999, A088218, A110555.

Table[Binomial[2n-1,n](-1)^n,{n,0,30}] (* Harvey P. Dale, Apr 01 2012 *)
a[ n_] := (-1)^n Binomial[ 2 n - 1, n] (* Michael Somos, May 21 2013 *)
a[ n_] := Binomial[ -n, -2 n] (* Michael Somos, May 21 2013 *)
a[ n_] := SeriesCoefficient[(1 + x)^-n, {x, 0, n}] (* Michael Somos, May 21 2013 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x)^-n + x * O(x^n), n))} /* Michael Somos, May 21 2013 */

a(n) = A110555(2*n,n), central terms in triangle A110555;
a(n) = A088218(n)*(-1)^n.
E.g.f.: E(x) = 1 - x/(G(0)+2*x) ; G(k) = (k+1)^2 - 2*x*(2*k+1) + 2*x*(2*k+3)*((k+1)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011
a(n) = coefficient of x^n of (1 / (1 + x))^n where 1 / (1 + x) is the g.f. of A033999. - Michael Somos, May 21 2013
HANKEL transform is A000027. HANKEL transform of a(n+1) is A033999(n+1). - Michael Somos, May 21 2013
E.g.f.: (1 + exp(-2*x) * BesselI(0,2*x)) / 2. - Ilya Gutkovskiy, Nov 03 2021
From Peter Bala, Feb 14 2024: (Start)
a(n) = binomial(-n, n).
O.g.f.: A(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) = 1/ (2 - c(-x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
The g.f. A(x) satisfies A(x/(1 - x)^2) = 1/(1 + x). (End)
a(n) ~ (-1)^n*2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 18 2024
D-finite with recurrence n*a(n) +2*(2*n-1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A010967 a(n) = binomial coefficient C(n,14).

Original entry on oeis.org

1, 15, 120, 680, 3060, 11628, 38760, 116280, 319770, 817190, 1961256, 4457400, 9657700, 20058300, 40116600, 77558760, 145422675, 265182525, 471435600, 818809200, 1391975640, 2319959400, 3796297200, 6107086800, 9669554100, 15084504396, 23206929840, 35240152720

Offset: 14

N. J. A. Sloane

nonn

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 14..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,14): n in [14..60]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,14),n=14..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,14],{n,14,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=14, 50, print1(binomial(n,14), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A110555(n+1,14). - Reinhard Zumkeller, Jul 27 2005
a(n+13) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)/14!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
G.f.: x^14/(1-x)^15. - Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009
a(n) = n/(n-14) * a(n-1), n > 14. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=14} 1/a(n) = 14/13.
Sum_{n>=14} (-1)^n/a(n) = A001787(14)*log(2) - A242091(14)/13! = 114688*log(2) - 102309709/1287 = 0.9404563356... (End)

Some formulas rewritten for the correct offset by R. J. Mathar, Jul 07 2009

A010968 a(n) = binomial(n,15).

Original entry on oeis.org

1, 16, 136, 816, 3876, 15504, 54264, 170544, 490314, 1307504, 3268760, 7726160, 17383860, 37442160, 77558760, 155117520, 300540195, 565722720, 1037158320, 1855967520, 3247943160, 5567902560, 9364199760, 15471286560, 25140840660, 40225345056, 63432274896

Offset: 15

N. J. A. Sloane

nonn

There are no primes in this sequence. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 15..1000
Milan Janjic, Two Enumerative Functions University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,15): n in [15..70]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,15),n=15..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,15],{n,15,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=15, 50, print1(binomial(n,15), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,15). - Reinhard Zumkeller, Jul 27 2005
a(n+14) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)(n+14)/15!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^15/(1-x)^16. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-15) * a(n-1), n > 15. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=15} 1/a(n) = 15/14.
Sum_{n>=15} (-1)^(n+1)/a(n) = A001787(15)*log(2) - A242091(15)/14! = 245760*log(2) - 1023103525/6006 = 0.9438350048... (End)

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009

A010969 a(n) = binomial(n,16).

Original entry on oeis.org

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2042975, 5311735, 13037895, 30421755, 67863915, 145422675, 300540195, 601080390, 1166803110, 2203961430, 4059928950, 7307872110, 12875774670, 22239974430, 37711260990, 62852101650, 103077446706, 166509721602

Offset: 16

N. J. A. Sloane

nonn

a(n) = A110555(n+1,16). - Reinhard Zumkeller, Jul 27 2005
Coordination sequence for 16-dimensional cyclotomic lattice Z[zeta_17].
In this sequence only 17 is prime. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 16..1000
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

[ Binomial(n,16): n in [16..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,16),n=16..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,16],{n,16,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=16, 50, print1(binomial(n,16), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n+15) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)(n+14)(n+15)/16!. - Artur Jasinski, Dec 02 2007
G.f.: x^16/(1-x)^17. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-16) * a(n-1), n > 16. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=16} 1/a(n) = 16/15.
Sum_{n>=16} (-1)^n/a(n) = A001787(16)*log(2) - A242091(16)/15! = 524288*log(2) - 16369704448/45045 = 0.9468480104... (End)

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009

cp's OEIS Frontend

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A001287 a(n) = binomial coefficient C(n,10).

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A071919 Number of monotone nondecreasing functions [n]->[m] for n >= 0, m >= 0, read by antidiagonals.

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A001288 a(n) = binomial(n,11).

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A010965 a(n) = binomial(n,12).

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A010966 a(n) = binomial(n,13).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A110556 a(n) = binomial(2n-1, n)(-1)^n.