A005712 -id:A005712 - cp's OEIS frontend

A005725 Quadrinomial coefficients.

1, 1, 3, 10, 31, 101, 336, 1128, 3823, 13051, 44803, 154518, 534964, 1858156, 6472168, 22597760, 79067375, 277164295, 973184313, 3422117190, 12049586631, 42478745781, 149915252028, 529606271560, 1872653175556, 6627147599476, 23471065878276, 83186110269928

Offset: 0

N. J. A. Sloane

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

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (1,3). - Joerg Arndt, Jul 05 2011

			For n=2, (x^3 + x^2 + x + 1)^2 = x^6 + 2x^5 + 3x^4 + 4x^3 + 3x^2 + 2x + 1, and the coefficient of x^n = x^2 is 3, so a(2) = 3. - _Michael B. Porter_, Aug 15 2016

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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..200
R. K. Guy, Letter to N. J. A. Sloane, 1987

Cf. A008287.

Column k=3 of A305161.

P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2+x^3)^n)[n+1]: n in [0..25] ]; // Bruno Berselli, Jul 05 2011

seq(add(binomial(n,2*k)*binomial(n,k), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
a := n -> add(binomial(n,j)*binomial(n,2*j),j=0..n): seq(a(n), n=1..25); # Zerinvary Lajos, Feb 12 2007
seq(coeff(series(RootOf((16*x^3+8*x^2+11*x-4)*A^3+(3-2*x)*A+1, A), x=0, n+1), x, n), n=0..30);  # Mark van Hoeij, Apr 30 2013

a[n_] := Coefficient[(1+x+x^2+x^3)^n, x^n]; a[0] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 15 2011 *)
Table[HypergeometricPFQ[{1/2 - n/2, -n, -n/2}, {1/2, 1}, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

quadrinomial(n,k):=coeff(expand((1+x+x^2+x^3)^n),x,k); makelist(quadrinomial(n,n),n,0,12); /* Emanuele Munarini, Mar 15 2011 */

a(n)=my(x='x); polcoeff((x^3+x^2+x+1)^n,n) \\ Charles R Greathouse IV, Feb 07 2017

from math import comb
def A005725(n): return sum((-1 if k&1 else 1)*comb(n,k)*comb((n-(k<<1)<<1)-1,n-(k<<2)) for k in range((n>>2)+1)) if n else 1 # Chai Wah Wu, Aug 09 2025

a(n) = Sum_{i+j+k=n, 0<=k<=j<=i<=n} C(n,i)*C(i,j)*C(j,k). - Benoit Cloitre, Jun 06 2004
G.f.: A(x) where (16*x^3+8*x^2+11*x-4)*A(x)^3+(3-2*x)*A(x)+1 = 0. - Mark van Hoeij, Apr 30 2013
Recurrence: 2*n*(2*n-1)*(13*n-19)*a(n) = (143*n^3 - 352*n^2 + 251*n - 54)*a(n-1) + 4*(n-1)*(26*n^2 - 51*n + 15)*a(n-2) + 16*(n-2)*(n-1)*(13*n-6)*a(n-3). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ sqrt((39+7*39^(2/3)/c+39^(1/3)*c)/156) * ((b+11+217/b)/12)^n/sqrt(Pi*n), where b = (6371+624*sqrt(78))^(1/3), c = (117+2*sqrt(78))^(1/3). - Vaclav Kotesovec, Aug 10 2013
a(n) = A008287(n, n). - Sean A. Irvine, Aug 15 2016
a(n) = hypergeom([1/2-n/2, -n, -n/2], [1/2, 1], -1). - Vladimir Reshetnikov, Oct 04 2016
From Peter Bala, Mar 31 2020: (Start)
a(n) = Sum_{k = 0..floor(n/4)} (-1)^k*C(n,k)*C(2*n-4*k-1,n-4*k).
a(p) == 1 (mod p^2) for any prime p >= 3. More generally, we may have a(p^k) == a(p^(k-1)) (mod p^(2*k)) for k >= 2 and any prime p >= 3.
The sequence defined by b(n) := [x^n] ( F(x)/F(-x) )^n, where F(x) = 1 + x + x^2 + x^3, may satisfy the stronger supercongruences b(p) == 2 (mod p^3) for prime p >= 5 (checked up to p = 499). (End)
a(n) = Sum_{k = 0..floor(n/2)} binomial(n,k)*binomial(n,2*k). - Peter Bala, Mar 16 2023

More terms from James Sellers, Jul 12 2000

A005721 Central quadrinomial coefficients.

Original entry on oeis.org

1, 4, 44, 580, 8092, 116304, 1703636, 25288120, 379061020, 5724954544, 86981744944, 1327977811076, 20356299454276, 313095240079600, 4829571309488760, 74683398325804080, 1157402982351003420, 17971185794898859248

Offset: 0

N. J. A. Sloane

Sum of squares of entries in the n-th row of triangle of quadrinomial coefficients A008287 (Pascal triangle of order 4). - Adi Dani, Jul 03 2011

Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n), see link. - Zagros Lalo, Sep 25 2018

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 601, 602.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..597 (first 101 terms from T. D. Noe)
Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23.
Adi Dani, Restricted compositions of natural numbers-section: Generalized Pascal triangle
R. K. Guy, editor, Western Number Theory Problems, 1985-12-21 & 23, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission. See Problem 85:02.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Zagros Lalo, Formula for the Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n).

Cf. A002426, A005190, A008287.

List([0..20],n->Sum([0..Int(3*n/4)],k->(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k))); # Muniru A Asiru, Sep 26 2018

[(&+[(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k): k in [0..Floor(3*n/4)]]): n in [0..30]]; // G. C. Greubel, Oct 06 2018

F := (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1); G := t/((t+1)^2*(6*t+t^2+1));
Ginv := RootOf(numer(G-x),t); series(eval(F,t=Ginv),x=0,20);
seq(coeff((1+x+x^2+x^3)^(2*n),x,3*n),n=0..50); # Robert Israel, Nov 01 2015

Table[Sum[(-1)^k*Binomial[2*n,k]*Binomial[5*n-4*k-1,3*n-4*k],{k,0,3*n/4}],{n,0,25}] (* Adi Dani, Jul 03 2011 *)
RecurrenceTable[{128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a[n-2] -8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a[n-1]+3*n*(3*n-2)*(3*n-1)*(5*n-6)*a[n]==0,
a[0]==1,a[1]==4},a,{n,0,5000}] (* Bradley Klee, Jun 25 2018 *)
a[n_] := a[n] = Sum[(2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!), {i, 0, n}, {j, n, 2*n}]; Table[a[n], {n, 0, 20}] (* Zagros Lalo, Sep 25 2018 *)

a(n)={local(v=Vec((1+x+x^2+x^3)^n));sum(k=1,#v, v[k]^2);}

a(n)=sum(k=0,3*n/4, (-1)^k*binomial(2*n,k)*binomial(5*n-4*k-1,3*n-4*k));

vector(30, n, n--; polcoeff((1+x+x^2+x^3)^(2*n), (6*n)>>1)) \\ Altug Alkan, Nov 01 2015

a(n) = A005190(2*n) = A008287(2*n, 3*n).
G.f.:  Let Z(x) be a solution of (-1+16*x)*(32*x-27)^2*Z^6+9*(-9+64*x)*(32*x-27)*Z^4+81*(80*x-27)*Z^2+729 = 0, with Z(0)=1. Compute a Puiseux series for Z(x) at x=0, then Z(x) in C[[x^(1/3)]].  Remove all non-integer powers of x.  The result is the generating function for A005721.  - Mark van Hoeij, Oct 29 2011
G.f.: F(G^(-1)(x)) where F(t) = (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1) and G(t) = t/((t+1)^2*(6*t+t^2+1)). - Mark van Hoeij, Oct 30 2011
From Bradley Klee, Jun 25 2018: (Start)
128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a(n-2) - 8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a(n-1) + 3*n*(3*n-2)*(3*n-1)*(5*n-6)*a(n) = 0.
G.f. G(x) satisfies a Picard-Fuchs type differential equation, 0 = Sum_{m=0..5, n=0..3} M_{m,n} x^m*(d^n/dx^n G(x)), with integer matrix:
M={{  24,     -6,      0,     0},
   {-768,   1488,    -54,     0},
   {6144, -16128,   2520,   -27},
   {   0,  55296, -29568,   896},
   {   0,      0,  49152, -7936},
   {   0,      0,      0,  8192}}(End)
a(n) = sum_{k=0..floor(3n/4)} (-1)^k binomial(2n,k) * binomial(5n-4k-1,3n-4k). - Muniru A Asiru, Sep 26 2018
a(n) = Sum_{i=0..n} Sum_{j=n..2n}(f); f= ( (2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!) ); f=0 for (3*n + i - 2*j)<0 or (j - 2*i)<0. See also formula in Links section. - Zagros Lalo, Sep 27 2018

A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n.

Original entry on oeis.org

3, 16, 51, 126, 266, 504, 882, 1452, 2277, 3432, 5005, 7098, 9828, 13328, 17748, 23256, 30039, 38304, 48279, 60214, 74382, 91080, 110630, 133380, 159705, 190008, 224721, 264306, 309256, 360096, 417384, 481712, 553707, 634032, 723387, 822510

Offset: 3

N. J. A. Sloane

If Y is a 3-subset of an n-set X then, for n>=7, a(n-4) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
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
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A005581, A005712, A005714-A005716, A111808.
Column m=5 of (1, 3) Pascal triangle A095660.
Cf. A000217, A055998.

[3*Binomial(n+2,5)-2*Binomial(n+1,5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012

A000574:=-(-3+2*z)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
seq(3*binomial(n+2,5)-2*binomial(n+1,5),n=3..100); # Robert Israel, Aug 04 2015
A000574 := n -> GegenbauerC(`if`(5A000574(n)), n=3..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(3-2*x)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)

x='x+O('x^50); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017

G.f.: x^3*(3-2*x)/(1-x)^6.
a(n) = 3*binomial(n+2,5) - 2*binomial(n+1,5).
a(n) = A111808(n,5) for n>4. - Reinhard Zumkeller, Aug 17 2005
a(n) = binomial(n+1, 4)*(n+12)/5 = 3*b(n-3)-2*b(n-4), with b(n)=binomial(n+5, 5); cf. A000389.
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). - Vincenzo Librandi, Jun 10 2012
a(n) = 3*binomial(n, 3) + 4*binomial(n, 4) + binomial(n, 5). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 5 if 5Peter Luschny, May 10 2016
a(n) = Sum_{i=1..n-1} A000217(i)*A055998(n-1-i). - Bruno Berselli, Mar 05 2018
E.g.f.: exp(x)*x^3*(60 + 20*x + x^2)/120. - Stefano Spezia, Jul 09 2023

More terms from Vladeta Jovovic, Oct 02 2000

A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314, 52624, 93093, 157950, 258570, 410346, 633726, 955434, 1409895, 2040885, 2903428, 4065963, 5612805, 7646925, 10293075, 13701285, 18050760, 23554206, 30462615, 39070540, 49721892, 62816292, 78816012, 98253540

Offset: 4

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000574, A005581, A005712, A005714, A005715, A111808.

I:=[1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[9]: n in [4..32] ]; // Bruno Berselli, Jun 17 2012

A005716:=-(6*z-9*z**2+3*z**3+1)/(z-1)**9; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005716 := n -> GegenbauerC(`if`(8A005716(n)), n=4..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(1+6*x-9*x^2+3*x^3)/(1-x)^9,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

a(n) = binomial(n+1, 5)*(n^2+23*n-84)*(n+10)/336, n >= 4.
G.f.: (x^4)*(1+6*x-9*x^2+3*x^3)/(1-x)^9. (Numerator polynomial is N3(8, x) from A063420).
a(n) = A027907(n, 8), n >= 4 (ninth column of trinomial coefficients).
a(n) = A111808(n,8) for n>7. - Reinhard Zumkeller, Aug 17 2005
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). Vincenzo Librandi, Jun 16 2012
a(n) = binomial(n,4) + 10*binomial(n,5) + 15*binomial(n,6) + 7*binomial(n,7) + binomial(n,8) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 8 if 8Peter Luschny, May 10 2016

More terms from Vladeta Jovovic, Oct 02 2000

A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

1, 10, 45, 141, 357, 784, 1554, 2850, 4917, 8074, 12727, 19383, 28665, 41328, 58276, 80580, 109497, 146490, 193249, 251713, 324093, 412896, 520950, 651430, 807885, 994266, 1214955, 1474795, 1779121, 2133792, 2545224, 3020424, 3567025, 4193322, 4908309, 5721717

Offset: 3

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000574, A005581, A005712, A005715-A005716, A111808.

I:=[1, 10, 45, 141, 357, 784, 1554]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // Bruno Berselli, Jun 17 2012

A005714:=-(1+3*z-4*z**2+z**3)/(z-1)**7; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005714 := n -> GegenbauerC(`if`(6A005714(n)), n=3..20); # Peter Luschny, May 10 2016

a[n_] := Coefficient[(1 + x + x^2)^n, x, 6]; Table[a[n], {n, 3, 35}]
CoefficientList[Series[(1+3*x-4*x^2+x^3)/(1-x)^7,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

a(n) = binomial(n, 3)*(n^3+18*n^2+17*n-120) /120.
G.f.: (x^3)*(1+3*x-4*x^2+x^3)/(1-x)^7. (Numerator polynomial is N3(6, x) from A063420).
a(n) = A027907(n, 6), n >= 3 (seventh column of trinomial coefficients).
a(n) = A111808(n,6) for n>5. - Reinhard Zumkeller, Aug 17 2005
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). Vincenzo Librandi, Jun 16 2012
a(n) = binomial(n,3) + 6*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 6 if 6Peter Luschny, May 10 2016
E.g.f.: exp(x)*x^3*(120 + 180*x + 30*x^2 + x^3)/720. - Stefano Spezia, Mar 28 2023

More terms from Vladeta Jovovic, Oct 02 2000

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 19, 10, 1, 1, 15, 45, 45, 15, 1, 1, 21, 90, 141, 90, 21, 1, 1, 28, 161, 357, 357, 161, 28, 1, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55

Offset: 1

Colin Mallows, Aug 23 2000

Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna, Feb 26 2005

Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2006

			Triangle begins:
  1;
  1,1;
  1,3,1;
  1,6,6,1;
  1,10,19,10,1;
  ...
Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3, 1;
  0, 1, 6, 6, 1;
  0, 1, 10, 19, 10, 1;
  0, 1, 15, 45, 45, 15, 1;
  0, 1, 21, 90, 141, 90, 21, 1;
  ... - _Philippe Deléham_, Mar 27 2014

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Oct 11 2011, after Paul Barry *)

PARI
```
T(n,k)=if(nPaul D. Hanna
```

T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna, Feb 26 2005
Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - Philippe Deléham, Oct 29 2006
From Paul Barry, Sep 28 2010: (Start)
G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).
E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).
T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - Philippe Deléham, Mar 27 2014

More terms from James Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005

A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

Original entry on oeis.org

3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535

Offset: 0

N. J. A. Sloane

If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - David Consiglio, Jr., Oct 27 2008
Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
R. K. Guy, Letter to N. J. A. Sloane, 1987.
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.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008287, A062745, A063421, A071675, A213548.

[(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)

a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.
G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n) = A062745(n, 4), n >= 2 (fifth column).
a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - Stefano Spezia, May 09 2024

Better description from Zerinvary Lajos, Dec 02 2005

A001919 Eighth column of quadrinomial coefficients.

Original entry on oeis.org

6, 40, 155, 456, 1128, 2472, 4950, 9240, 16302, 27456, 44473, 69680, 106080, 157488, 228684, 325584, 455430, 627000, 850839, 1139512, 1507880, 1973400, 2556450, 3280680, 4173390, 5265936, 6594165, 8198880, 10126336, 12428768, 15164952, 18400800, 22209990

Offset: 3

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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 = 3..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

seq(n*(n^2-1)*(n^2-4)*(n^2+21*n+180)/5040,n=3..34); # Emeric Deutsch, Jan 27 2005
A001919:=(3*z**2-8*z+6)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*(n^2 - 1)*(n^2 - 4)*(n^2 + 21*n + 180)/5040, {n, 3, 50}] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{6,40,155,456,1128,2472,4950,9240},40] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = A008287(n, 7) = binomial(n+2, 5)*(n^2+21*n+180 )/42, n >= 3.
G.f.: (x^3)*(6-8*x+3*x^2 )/(1-x)^8. Numerator polynomial is N4(7, x) from array A063421.
a(n) = n(n^2-1)(n^2-4)(n^2+21n+180)/5040. - Emeric Deutsch, Jan 27 2005
a(n) = 6*C(n,3) + 16*C(n,4) + 15*C(n,5) + 6*C(n,6) + C(n,7) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(3)=6, a(4)=40, a(5)=155, a(6)=456, a(7)=1128, a(8)=2472, a(9)=4950, a(10)=9240, a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - Harvey P. Dale, Mar 27 2013

More terms from Emeric Deutsch, Jan 27 2005

A005715 Coefficient of x^7 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

4, 30, 126, 393, 1016, 2304, 4740, 9042, 16236, 27742, 45474, 71955, 110448, 165104, 241128, 344964, 484500, 669294, 910822, 1222749, 1621224, 2125200, 2756780, 3541590, 4509180, 5693454, 7133130, 8872231, 10960608, 13454496

Offset: 4

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000574, A005581, A005712, A005714, A005716, A111808.

I:=[4, 30, 126, 393, 1016, 2304, 4740, 9042]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[8]: n in [4..33] ]; // Bruno Berselli, Jun 17 2012

A005715:=(z-2)*(z**2-2)/(z-1)**8; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005715 := n -> GegenbauerC(`if`(7A005715(n)), n=4..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(x-2)*(x^2-2)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

a(n) = binomial(n, 4)*(n^3+27*n^2+116*n-120)/210, n >= 4.
G.f.: (x^4)*(x-2)*(x^2-2)/(1-x)^8. (Numerator polynomial is N3(7, x) from A063420).
a(n) = A027907(n, 7), n >= 4 (eighth column of trinomial coefficients).
a(n) = A111808(n,7) for n>6. - Reinhard Zumkeller, Aug 17 2005
a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). Vincenzo Librandi, Jun 16 2012
a(n) = 4*binomial(n,4) + 10*binomial(n,5) + 6*binomial(n,6) + binomial(n,7) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 7 if 7Peter Luschny, May 10 2016

More terms from Vladeta Jovovic, Oct 02 2000

A005722 a(n) = (prime(n) - 1)^2.

Original entry on oeis.org

1, 4, 16, 36, 100, 144, 256, 324, 484, 784, 900, 1296, 1600, 1764, 2116, 2704, 3364, 3600, 4356, 4900, 5184, 6084, 6724, 7744, 9216, 10000, 10404, 11236, 11664, 12544, 15876, 16900, 18496, 19044, 21904, 22500, 24336, 26244, 27556, 29584, 31684, 32400, 36100

Offset: 1

Scorpion(AT)aol.com

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Richard K. Guy, Letter to N. J. A. Sloane, 1987.

Cf. A001248, A005597, A006093, A065485, A086242, A095874, A192134.

[(p-1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

A005722:=n->(ithprime(n)-1)^2; seq(A005722(n), n=1..40); # Wesley Ivan Hurt, Mar 27 2014

Table[(Prime[n] - 1)^2, {n, 40}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = (prime(n) - 1)^2; \\ Michel Marcus, Nov 09 2020

a(n) = A192134(A095874(A001248(n))) - 1. - Reinhard Zumkeller, Jun 26 2011
a(n) = A006093(n)^2. - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = A086242. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065485.
Product_{n>=2} (1 - 1/a(n)) = A005597. (End)

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A005725 Quadrinomial coefficients.

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A005721 Central quadrinomial coefficients.

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A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n.

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A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n.

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A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n.

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A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).