A008288 -id:A008288 - cp's OEIS frontend

A249938 E.g.f.: Sum_{n>=0} exp(n^2*x) / 2^(n+1).

1, 3, 75, 4683, 545835, 102247563, 28091567595, 10641342970443, 5315654681981355, 3385534663256845323, 2677687796244384203115, 2574844419803190384544203, 2958279121074145472650648875, 4002225759844168492486127539083, 6297562064950066033518373935334635, 11403568794011880483742464196184901963

Offset: 0

Paul D. Hanna, Nov 20 2014

nonn

a(n) == 3 (mod 72) for n>0.
Conjectures from Federico Provvedi, Nov 07 2020: (Start)
For n>1, a(n+1) - a(n) == 0 (mod m) if and only if m divides 288.
This sequence is a periodic sequence modulo m, and if m is the k-th prime, the periods of {a(n)} over k-th prime is the sequence of the number of nonzero quadratic residues modulo k-th prime, for all k>0.
Example: k=9, m = prime(9) = 23, for n>0, {a(n) mod 23} generates a period of 11 elements {3, 6, 14, 22, 5, 3, 10, 2, 4, 5, 0}, hence A130290(9) = 11
(End)

			E.g.f.: A(x) = 1 + 3*x + 75*x^2/2! + 4683*x^3/3! + 545835*x^4/4! +...
where the e.g.f. equals the infinite series:
A(x) = 1/2 + exp(x)/2^2 + exp(4*x)/2^3 + exp(9*x)/2^4 + exp(16*x)/2^5 + exp(25*x)/2^6 + exp(36*x)/2^7 + exp(49*x)/2^8 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Kwang-Wu Chen, Multinomial Sum Formulas of Multiple Zeta Values, arXiv:1704.05636 [math.NT], 2017.

Cf. A249941, A247082, A250914, A250915, A000670.

Table[Sum[k! * StirlingS2[2*n, k],{k,0,2*n}],{n,0,20}] (* Vaclav Kotesovec, May 04 2015 *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; a[n_] := Fubini[2n, 1]; a[0] = 1; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Mar 30 2016 *)
Table[-PolyLog[-2*n, 2] / 2, {n, 0, 48}] (* Federico Provvedi, Nov 07 2020 *)
HurwitzLerchPhi[1/2, -2*Range[0,48], 0] / 2 (* Federico Provvedi, Nov 11 2020 *)
-HurwitzLerchPhi[2, -2*Range[0, 48], 1] (*Federico Provvedi,Nov 11 2020*)

/* E.g.f.: Sum_{n>=0} exp(n^2*x)/2^(n+1) */
\p100 \\ set precision
{a(n) = round( n!*polcoeff(sum(m=0, 600, exp(m^2*x +x*O(x^n))/2^(m+1)*1.), n) )}
for(n=0, 20, print1(a(n), ", "))

/* E.g.f.: (2 - cosh(x)) / (5 - 4*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (2 - cosh(X)) / (5 - 4*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = sum(k=0, 2*n, k! * Stirling2(2*n, k) )}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: (2 - cosh(x)) / (5 - 4*cosh(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.
a(n) = Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>=0.
a(n) = A000670(2*n), where A000670 is the Fubini numbers.
a(n) ~ (2*n)! / (2 * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 04 2015
a(n) = Sum_{p=1..k, q=1..k} Stirling2(k,p)*Stirling2(k,q)*p!*q!*A008288(p, q) for n>1, where A008288 are the Delannoy numbers. See Chen link. - Michel Marcus, Apr 20 2017
a(n) = Sum_{k>=0} k^(2*n) / 2^(k + 1). - Ilya Gutkovskiy, Dec 19 2019
a(n) = -Polylog(-2*n, 2) / 2. - Federico Provvedi, Nov 07 2020
a(n) = Phi(1/2, -2*n, 0), where Phi(z,s,a) is the Hurwitz-Lerch Zeta transcendental function. - Federico Provvedi, Nov 11 2020

A001848 Crystal ball sequence for 6-dimensional cubic lattice.

Original entry on oeis.org

1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, 24957661, 31388293, 39146185, 48442297, 59511829, 72616013, 88043969

Offset: 0

N. J. A. Sloane

Number of nodes of degree 12 in virtual, optimal chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
Equals binomial transform of [1, 12, 60, 160, 240, 192, 64, 0, 0, 0, ...] where (1, 12, 60, 160, 240, 192, 64) = row 6 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008
a(n) is the number of points in Z^6 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 6 from any given point. - Shel Kaphan, Jan 02 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
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 = 0..1000
D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001847, A013609.
Cf. A240876.
Row/column 6 of A008288.

for n from 1 to k do eval(4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1); od;
A001848:=-(z+1)**6/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[-(z + 1)^6/(z - 1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

G.f.: (1+x)^6 /(1-x)^7.
a(n) = (4/45)*n^6 + (4/15)*n^5 + (14/9)*n^4 + (8/3)*n^3 + (196/45)*n^2 + (46/15)*n + 1. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
a(n) = Sum_{k=0..min(6,n)} 2^k * binomial(6,k)* binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 37/60 = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6). - Peter Bala, Mar 23 2024

A047662 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k)=a(n-1,k-1)+a(n-1,k)+a(n,k-1)+1.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 31, 20, 5, 6, 30, 64, 64, 30, 6, 7, 42, 115, 160, 115, 42, 7, 8, 56, 188, 340, 340, 188, 56, 8, 9, 72, 287, 644, 841, 644, 287, 72, 9, 10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10, 11, 110, 579, 1824, 3591

Offset: 1

Don Knuth, N. J. A. Sloane

			The array begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, ...
3, 12, 31, 64, 115, 188, 287, 416, 579, 780, 1023, 1312, ...
4, 20, 64, 160, 340, 644, 1120, 1824, 2820, 4180, 5984, 8320, ...
5, 30, 115, 340, 841, 1826, 3591, 6536, 11181, 18182, 28347, 42652, ...
6, 42, 188, 644, 1826, 4494, 9912, 20040, 37758, 67122, 113652, 184652, ...
7, 56, 287, 1120, 3591, 9912, 24319, 54272, 112071, 216952, 397727, 696032, ...
8, 72, 416, 1824, 6536, 20040, 54272, 132864, 299208, 628232, 1242912, 2336672, ...
...
The first few antidiagonals are:
1,
2, 2,
3, 6, 3,
4, 12, 12, 4,
5, 20, 31, 20, 5,
6, 30, 64, 64, 30, 6,
7, 42, 115, 160, 115, 42, 7,
8, 56, 188, 340, 340, 188, 56, 8,
9, 72, 287, 644, 841, 644, 287, 72, 9,
10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10,
...

Vincenzo Librandi, Rows n = 1..100, flattened
M. L. Fredman, The complexity of maintaining an array and its partial sums, J. Assoc. Comp. Machin., 29 (1982), 250-260.
D. E. Knuth and N. J. A. Sloane, Correspondence, December 1999
Matthew Roughan, Surreal Birthdays and Their Arithmetic, arXiv:1810.10373 [math.HO], 2018.

Rows give A037237, 4*A006007, A047661, A047663, A047664, main diagonal is A047665 (see also A001850).

A064861 Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 8, 4, 1, 6, 13, 12, 4, 1, 8, 25, 38, 28, 8, 1, 9, 33, 63, 66, 36, 8, 1, 11, 51, 129, 192, 168, 80, 16, 1, 12, 62, 180, 321, 360, 248, 96, 16, 1, 14, 86, 304, 681, 1002, 968, 592, 208, 32, 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32, 1, 17

Offset: 0

Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

When A064861 is regarded as a triangle read by rows, this is [1,0,-1,0,0,0,0,0,0,...] DELTA [2,-1,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 14 2008

			Table begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  2,  3,  5,   6,   8,  9, 11, ...
  2,  8, 13,  25,  33, 51, ...
  4, 12, 38,  63, 129, ...
  4, 28, 66, 192, ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
C. de Jesús Pita Ruiz Velasco, Convolution and Sulanke Numbers, JIS 13 (2010) 10.1.8.
R. A. Sulanke, Problem 10894, Amer. Math. Monthly 108, (2001), p. 770.

Cf. central Delannoy numbers a(n,n) = A001850(n), Delannoy numbers (same main diagonal): a(n,n) = A008288(n,n), a(n-1,n)=A002003(n), a(n,n+1)=A002002(n), a(n,1)=A058582(n), apparently a(n,n+2)=A050151(n).

a064861 n k = a064861_tabl !! n !! k
a064861_row n = a064861_tabl !! n
a064861_tabl = map fst $ iterate f ([1], 2) where
f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 3 - z)
-- Reinhard Zumkeller, May 01 2014

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else procname(n,k-1)+(3/2-1/2*(-1)^(n+k))*procname(n-1,k); fi; end;
seq(seq(A064861(i,j-i),i=1..j-1),j=1..19);

max = 12; se = Series[(1 + 2*x + y*x)/(1 - 2*x^2 - y^2*x^2 - 3*y*x^2), {x, 0, max}, {y, 0, max}]; cc = CoefficientList[se, {x, y}]; Flatten[ Table[ cc[[n, k]], {n, 1, max}, {k, n, 1, -1}]] (* Jean-François Alcover, Oct 21 2011, after g.f. *)

a(n,m)=if(n<0 || m<0,0,polcoeff(polcoeff((1+2*x+y*x)/(1-2*x^2-y^2*x^2-3*y*x^2)+O(x^(n+m+1)),n+m),m))

G.f.: Sum_{m>=0} Sum_{n>=0} a_{m, n}*t^m*s^n = A(t,s) = (1+2*t+s)/(1-2*t^2-s^2-3*s*t).

A101164 Triangle read by rows: Delannoy numbers minus binomial coefficients.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 15, 15, 4, 0, 0, 5, 26, 43, 26, 5, 0, 0, 6, 40, 94, 94, 40, 6, 0, 0, 7, 57, 175, 251, 175, 57, 7, 0, 0, 8, 77, 293, 555, 555, 293, 77, 8, 0, 0, 9, 100, 455, 1079, 1431, 1079, 455, 100, 9, 0, 0, 10, 126, 668, 1911, 3191, 3191, 1911, 668, 126, 10, 0

Offset: 0

Reinhard Zumkeller, Dec 03 2004

			Triangle begins as:
  0
  0, 0;
  0, 1,  0;
  0, 2,  2,   0;
  0, 3,  7,   3,   0;
  0, 4, 15,  15,   4,   0;
  0, 5, 26,  43,  26,   5,  0;
  0, 6, 40,  94,  94,  40,  6, 0;
  0, 7, 57, 175, 251, 175, 57, 7, 0;

Reinhard Zumkeller, Rows n = 0..100 of table, flattened
Eric Weisstein's World of Mathematics, Delannoy Number
Eric Weisstein's World of Mathematics, Binomial Coefficient
Index entries for triangles and arrays related to Pascal's triangle

Cf. A001477, A005449, A008288, A094706, A101165, A101166, A225413.

a101164 n k = a101164_tabl !! n !! k
a101164_row n = a101164_tabl !! n
a101164_tabl = zipWith (zipWith (-)) a008288_tabl a007318_tabl
-- Reinhard Zumkeller, Jul 30 2013

A101164:= func< n,k | (&+[Binomial(n-k,j)*Binomial(k,j)*2^j: j in [0..n-k]]) - Binomial(n,k) >;
[A101164(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 17 2021

T[n_, k_]:= Hypergeometric2F1[-k, k-n, 1, 2] - Binomial[n, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2021 *)

def T(n,k): return simplify(hypergeometric([-n+k, -k], [1], 2)) - binomial(n,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 17 2021

T(n, k) = A008288(n, k) - binomial(n, k), 0<=k<=n, where binomial=A007318.
T(n,2) = A005449(n-2) for n>1;
T(n,3) = A101165(n-3) for n>2;
T(n,4) = A101166(n-4) for n>3;
Sum_{k=0..n} T(n, k) = A094706(n).
From G. C. Greubel, Sep 17 2021: (Start)
T(n, k) = Sum_{j=0..n-k} binomial(n-k, j)*binomial(k, j)*2^j - binomial(n,k).
T(n, 1) = n-1, n > 0. (End)

A102413 Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 16, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 48, 76, 48, 12, 1, 1, 14, 70, 154, 154, 70, 14, 1, 1, 16, 96, 272, 384, 272, 96, 16, 1, 1, 18, 126, 438, 810, 810, 438, 126, 18, 1, 1, 20, 160, 660, 1520, 2004, 1520, 660, 160, 20, 1, 1, 22, 198, 946, 2618, 4334, 4334, 2618, 946, 198, 22, 1

Offset: 0

Emeric Deutsch, Jan 07 2005

The n-sunlet graph is the corona C'(n) of the cycle graph C(n) and the complete graph K(1); in other words, C'(n) is the graph constructed from C(n) to which for each vertex v a new vertex v' and the edge vv' is added.
Row n contains n+1 terms. Row sums yield A099425. T(n,k) = T(n,n-k).
For n > 2: same recurrence as A008288 and A128966. - Reinhard Zumkeller, Apr 15 2014

			T(3,2) = 6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC}, {Bb,AC}, {Cc,AB}, {Aa,Bb}, {Aa,Cc} and {Bb,Cc}.
The triangle starts:
  1;
  1, 1;
  1, 4,  1;
  1, 6,  6, 1;
  1, 8, 16, 8, 1;
From _Eric W. Weisstein_, Apr 03 2018: (Start)
Rows as polynomials:
  1
  1 +    x,
  1 +  4*x +    x^2,
  1 +  6*x +  6*x^2 +    x^3,
  1 +  8*x + 16*x^2 +  8*x^3 +    x^4,
  1 + 10*x + 30*x^2 + 30*x^3 + 10*x^4 + x^5,
  ... (End)

J. L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, Boca Raton, 2004, p. 894.
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969, p. 167.

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Frédéric Bihan, Francisco Santos, and Pierre-Jean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018.
A. F. Horadam, Chebyshev and Pell connections, Fib. Quart. 43 (2) (2005) 108-121, table (6.11)
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Sunlet Graph

Cf. A002203 or A099425 (row sums), A006318, A008288.

Cf. A241023 (central terms).

a102413 n k = a102413_tabl !! n !! k
a102413_row n = a102413_tabl !! n
a102413_tabl = [1] : [1,1] : f [2] [1,1] where
   f us vs = ws : f vs ws where
             ws = zipWith3 (((+) .) . (+))
                  ([0] ++ us ++ [0]) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 15 2014

G:=(1+t*z^2)/(1-(1+t)*z-t*z^2): Gser:=simplify(series(G,z=0,38)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od:for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

CoefficientList[Table[2^-n ((1 + x - Sqrt[1 + x (6 + x)])^n + (1 + x + Sqrt[1 + x (6 + x)])^n), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
LinearRecurrence[{1 + x, x}, {1, 1 + x, 1 + 4 x + x^2}, 10] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
Join[{1}, CoefficientList[CoefficientList[Series[(-1 - x - 2 x z)/(-1 + z + x z + x z^2), {z, 0, 10}], z], x]] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

G.f.: G(t,z) = (1 + t*z^2) / (1 - (1+t)*z - t*z^2).
For n > 2: T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), 0 < k < n. - Reinhard Zumkeller, Apr 15 2014 (corrected by Andrew Woods, Dec 08 2014)
From Peter Bala, Jun 25 2015: (Start)
The n-th row polynomial R(n, t) = [z^n] F(z, t)^n, where F(z, t) = 1/2*( 1 + (1 + t)*z + sqrt(1 + 2*(1 + t)*z + (1 + 6*t + t^2)*z^2) ).
exp( Sum_{n >= 1} R(n, t)*z^n/n ) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ... is the o.g.f for A008288 read as a triangular array. (End)
From Peter Bala, Aug 01 2024: (Start)
T(n, k) = A008288(n-k, k) + A008288(n-k-1, k-1) (Bihan et al., Proposition 6.6).
T(n, k) = 1 if n = 0 or k = n, else for 1 <= k <= n-1, T(n, k) = Sum_{j = 0..min(n-k, k)} (2^j)*(binomial(n-k, j)*binomial(k, j) + binomial(n-k-1, j)*binomial(k-1, j)).
Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. The signed n-th row polynomial R(n, -x) = 1/S(x)^n + (x*S(x))^n. (End)

Row 0 in polynomials and Mathematica programs added by Georg Fischer, Apr 01 2019

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

Original entry on oeis.org

1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601

Offset: 0

Paul D. Hanna, Jan 10 2012

nonn

a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...
where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...
The last digit of the terms in this sequence seems to be either a '1' or a '5':
by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;
if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..625

Cf. A204062, A026933, A001333, A023729.

{A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)),n)}

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.
a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 9, 1, 1, 25, 41, 13, 1, 1, 41, 129, 85, 17, 1, 1, 61, 321, 377, 145, 21, 1, 1, 85, 681, 1289, 833, 221, 25, 1, 1, 113, 1289, 3653, 3649, 1561, 313, 29, 1, 1, 145, 2241, 8989, 13073, 8361, 2625, 421, 33, 1, 1, 181, 3649, 19825, 40081, 36365, 16641, 4089, 545, 37, 1

Offset: 0

Paul Barry, Feb 07 2006, Oct 22 2006

Row sums are A099463(n+1). Diagonal sums are A116404.

Triangle formed of even-numbered columns of the Delannoy triangle A008288. - Philippe Deléham, Mar 11 2013

			Triangle begins
  1;
  1,  1;
  1,  5,   1;
  1, 13,   9,   1;
  1, 25,  41,  13,   1;
  1, 41, 129,  85,  17,  1;
  1, 61, 321, 377, 145, 21, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A008288, A099463 (row sums), A116404 (diagonal sums), A184883.

T:= func< n, k | (&+[Binomial(2*k, j)*Binomial(n-k, j)*2^j: j in [0..n-k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2021

T := (n,k) -> hypergeom([-2*k, k-n], [1], 2);
seq(seq(round(evalf(T(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 16 2014

T[n_, k_] := Hypergeometric2F1[-2k, k-n, 1, 2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

def A114123(n,k): return round( hypergeometric([-2*k, k-n], [1], 2) )
flatten([[A114123(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 20 2021

T(n, k) = Sum_{j=0..n} C(2*k,n-k-j)*C(n-k,j)*2^(n-k-j).
T(n, k) = Sum_{j=0..n-k} C(2*k,j)*C(n-k,j)*2^j.
Sum_{k=0..n} T(n, k) = A099463(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A116404(n).
T(n, k) = hypergeom([-2*k, k-n], [1], 2). - Peter Luschny, Sep 16 2014
T(n, n-k) = A184883(n, k). - G. C. Greubel, Nov 20 2021

A190666 Number of walks from (0,0) to (n+3,n) which take steps from {E, N, NE}.

Original entry on oeis.org

1, 9, 61, 377, 2241, 13073, 75517, 433905, 2485825, 14218905, 81270333, 464387817, 2653649025, 15167050785, 86716873725, 495998874593, 2838240338817, 16248650965289, 93065296937533, 533285164334169, 3057236753252161, 17534423944871729, 100609937775369981

Offset: 0

Shanzhen Gao, May 25 2011

+-3-diagonal of A008288 as a square array. - Shel Kaphan, Jan 07 2023

S. Gao, H. Niederhausen, Counting New Lattice Paths and Walks with Several Step Vectors (submitted to Congr. Numer.). - Shanzhen Gao, May 25 2011

Alois P. Heinz, Table of n, a(n) for n = 0..400
Shanzhen Gao and Keh-Hsun Chen, Counting Lattice Paths and Walks with Several Step Vectors, FCS 2014.

Cf. A001850, A008288, A113139, A002002, A026002.

b:= proc(i, j) option remember;
      if i<0 or j<0 then 0
    elif i=0 and j=0 then 1
    else b(i-1, j) +b(i, j-1) +b(i-1, j-1)
      fi
    end:
a:= n-> b(n+3, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2011

b[i_, j_] /; i<0 || j<0 = 0; b[0, 0] = 1; b[i_, j_]:= b[i, j]= b[i-1, j] + b[i, j-1] + b[i-1, j-1]; a[n_] := b[n+3, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 01 2011, after Maple prog. *)
CoefficientList[Series[(-1+3*x-x^2+(1-6*x+6*x^2-x^3)/Sqrt[x^2-6*x+1])/(2*x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
Table[(-1)^n Hypergeometric2F1[-n, n+4, 1, 2], {n,0,22}] (* Peter Luschny, Mar 02 2017 *)

a(n) = Sum_{k=0..n} C(n,k) * C(n+k+3,k+3) = A113139 (n+3,3). - Alois P. Heinz, Jun 01 2011
G.f.: (-1 + 3*x - x^2 + (1 - 6*x + 6*x^2 - x^3)/sqrt(x^2 - 6*x + 1))/(2*x^3). - Alois P. Heinz, Jun 03 2011
Recurrence: n*(n+3)*a(n) = (5*n^2 + 15*n + 16)*a(n-1) + (5*n^2 - 5*n + 6)*a(n-2) - (n-2)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(1632 + 1154*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
From Peter Bala, Mar 02 2017: (Start)
a(n) = (1/2^(n+1))*Sum_{k >= 3} (1/2^k)*binomial(n+k, k)*binomial(n+k, n+3).
a(n) = (-1)^n*Sum_{k = 0..n} (-2)^k*binomial(n,k) * binomial(n+k+3,k).
n*(n+3)*(2*n + 1)*a(n) = 6*(n+1)*(2*n^2 + 4*n + 3)*a(n-1) - (n-1)*(n+2)*(2*n + 3)*a(n-2) with a(0) = 1 and a(1) = 9. (End)
a(n) = (-1)^n*hypergeom([-n, n+4], [1], 2). - Peter Luschny, Mar 02 2017

A328300 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (0,k,n-k), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 26, 15, 1, 1, 31, 82, 82, 31, 1, 1, 63, 237, 343, 237, 63, 1, 1, 127, 651, 1257, 1257, 651, 127, 1, 1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1, 1, 511, 4494, 13669, 22411, 22411, 13669, 4494, 511, 1, 1, 1023, 11485, 42279, 83680, 103730, 83680, 42279, 11485, 1023, 1

Offset: 0

Alois P. Heinz, Oct 11 2019

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,    1;
  1,  15,   26,   15,    1;
  1,  31,   82,   82,   31,    1;
  1,  63,  237,  343,  237,   63,    1;
  1, 127,  651, 1257, 1257,  651,  127,   1;
  1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Columns k=0-1 give: A000012, A000225.
Row sums give A328296.
T(2n,n) gives A328269.
T(n,floor(n/2)) gives A328280.
Cf. A007318, A008288, A091533, A328297, A328347.

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
T:= (n, k)-> b(sort([0, k, n-k])):
seq(seq(T(n, k), k=0..n), n=0..12);

b[l_List] := b[l] = If[l[[-1]] == 0, 1, Function[r, Sum[Sum[Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {k, r}], {j, r}], {i, r}]][{-1, 0, 1}]];
T[n_, k_] := b[Sort[{0, k, n - k}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 10 2020, after Maple *)

T(n,k) = T(n,n-k).

cp's OEIS Frontend

A249938 E.g.f.: Sum_{n>=0} exp(n^2*x) / 2^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A001848 Crystal ball sequence for 6-dimensional cubic lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A047662 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k)=a(n-1,k-1)+a(n-1,k)+a(n,k-1)+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A064861 Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A101164 Triangle read by rows: Delannoy numbers minus binomial coefficients.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Sage

Formula

A102413 Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions