A008287 -id:A008287 - cp's OEIS frontend

A330509 Triangle read by rows: T(n,k) is the number of 4-ary strings of length n with k indispensable digits, with 0 <= k <= n.

1, 1, 3, 1, 9, 6, 1, 19, 34, 10, 1, 34, 115, 91, 15, 1, 55, 301, 445, 201, 21, 1, 83, 672, 1582, 1338, 392, 28, 1, 119, 1344, 4600, 6174, 3410, 700, 36, 1, 164, 2478, 11623, 22548, 19784, 7723, 1170, 45, 1, 219, 4290, 26452, 69834, 88428, 55009, 15999, 1857, 55

Offset: 0

Ji Young Choi, Dec 16 2019

A digit in a string is called indispensable if it is greater than the following digit or equal to the following digits which are eventually greater than the following digit. We also assume that there is an invisible digit 0 at the end of any string. For example, in 7233355548, the digits 7, 5, 5, 5, and 8 are indispensable.

T(n, k) is also the number of integers m where the length of the base-4 representation of m is n and the digit sum of the base-4 representation of 3m is 3k.

			Triangle begins
  1;
  1,   3;
  1,   9,   6;
  1,  19,  34,  10;
  1,  34, 115,  91,  15;
  1,  55, 301, 445, 201,  21;
  ...
There is 1 string (00) of length 2 with 0 indispensable digits.
There are 9 strings (01, 02, 03, 10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digit.
There are 6 strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits.
Hence T(2,0)=1, T(2,1)=9, T(2,2)=6.

J. Y. Choi, Indispensable digits for digit sums, Notes Number Theory Discrete Math 25 (2019), pp. 40-48.
J. Y. Choi, Digit sums generalizing binomial coefficients, J. Integer Seq. 22 (2019), Article 19.8.3.

Cf. A008287, A330381.

Table[Total@ Map[Sum[Binomial[n, i] Binomial[n, # - 2 i], {i, 0, #/2}] &, 3 k + {-2, -1, 0}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 23 2019, after Jean-François Alcover at A008287 *)

A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
T(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);

T(n, k) = A008287(n, 3k-2) + A008287(n, 3k-1) + A008287(n, 3k).

A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 15, 4, 1, 42, 91, 34, 5, 1, 132, 603, 364, 65, 6, 1, 429, 4213, 4269, 1085, 111, 7, 1, 1430, 30537, 52844, 19845, 2666, 175, 8, 1, 4862, 227475, 679172, 383251, 70146, 5719, 260, 9, 1, 16796, 1730787, 8976188, 7687615, 1949156, 204687, 11096, 369, 10, 1

Offset: 1

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |  1    2     3      4      5
----+----------------------------
  1 |  1    1     1      1      1 ...
  2 |  2    3     4      5      6 ...
  3 |  5   15    34     65    111 ...
  4 | 14   91   364   1085   2666 ...
  5 | 42  603  4269  19845  70146 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000012 (n=1), A220892 (n=4).
Cf. A000108 (s=1), A099251 (s=2), A264607 (s=3).
Cf. A000027, A007318, A008287, A027907, A035343, A063260.
Cf. A349933.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]-T[2n,s n+1,s]; Flatten[Table[A[n-s+1,s],{n,10},{s,n}]]

T(n, k, s) = polcoef((sum(i=0, s, x^i))^n, k);
A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s); \\ Michel Marcus, Dec 10 2021

A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
A(2, n) = A000027(n+1).
A(3, n) = A006003(n+1).

A177960 Numbers of the form A001317(t), excluding those at places of the form t=m*(2^k-1), m>=0, k>=2.

Original entry on oeis.org

3, 5, 17, 51, 257, 1285, 3855, 13107, 65537, 196611, 983055, 1114129, 5570645, 16711935, 50529027, 84215045, 858993459, 4294967297, 21474836485, 219043332147, 365072220245, 1103806595329, 3311419785987

Offset: 1

Vladimir Shevelev, Dec 24 2010

nonn

m-nomial (m>=2) coefficients are coefficients of the polynomial (1+x+...+x^(m-1))^n (n>=0), see A007318 (m=2), A027907 (m=3), A008287 (m=4), A035343 (m=5) etc. For k>=1, consider the triangle of 2^k-nomial coefficients, each entry reduced mod 2, and convert each row of the reduced triangle to a single number by interpreting the sequence of bits as binary representation of a number. This defines sequences A001317 (k=1), A177882 (k=2), A177897 (k=3), etc. The current sequence lists terms of A001317 which are not derived from any of the sequences for k >=2, not from 4-nomial, not from 8-nomial, not from 16-nomial etc.

Conjecture: If for every m>=2, to consider triangle of m-nomial coefficients mod 2 converted to decimal, then the sequence lists terms of A001317 which are not in the union of other sequences for m=3 (A038184), 4 (A177882), 5, 6,...

Cf. A001317, A177882, A177897, A027907, A008287.

Denote by B(n) the number of terms of the sequence among the first n terms of A001317. Then lim_{n->infinity} B(n)/ = Product_{prime p>=2} (1 - 1/(2^p-1)) = A184085.

A178619 Triangle T(n,k) with the coefficient of [x^k] of the series (1-x)^(n+1)* sum_{j>=0} binomial(n + 4j, 4j)*x^j in row n, column k.

Original entry on oeis.org

1, 1, 3, 1, 12, 3, 1, 31, 31, 1, 1, 65, 155, 35, 1, 120, 546, 336, 21, 1, 203, 1554, 1918, 413, 7, 1, 322, 3823, 8092, 3823, 322, 1, 1, 486, 8451, 27876, 23607, 4950, 165, 1, 705, 17205, 82885, 112035, 44803, 4455, 55, 1, 990, 32802, 220198, 440484, 291258

Offset: 0

Roger L. Bagula, May 30 2010

Every fourth row is symmetrical.
Row sums are 4^n.
3*k instead of 4*k in the binomial() gives A178618.

			1;
1, 3;
1, 12, 3;
1, 31, 31, 1;
1, 65, 155, 35;
1, 120, 546, 336, 21;
1, 203, 1554, 1918, 413, 7;
1, 322, 3823, 8092, 3823, 322, 1;
1, 486, 8451, 27876, 23607, 4950, 165;
1, 705, 17205, 82885, 112035, 44803, 4455, 55;
1, 990, 32802, 220198, 440484, 291258, 59950, 2882, 11;

Cf. A178618, A008287.

A178619 := proc(n,k)
    (1-x)^(n+1)*add( binomial(n+4*j,4*j)*x^j,j=0..n+1) ;
    coeftayl(%,x=0,k) ;
end proc:
seq(seq(A178619(n,k),k=0..n),n=0..8) ; # R. J. Mathar, Nov 05 2012

p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 4*k, 4*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]

A265644 Triangle read by rows: T(n,m) is the number of quaternary words of length n with m strictly increasing runs (0 <= m <= n).

Original entry on oeis.org

1, 0, 4, 0, 6, 10, 0, 4, 40, 20, 0, 1, 65, 155, 35, 0, 0, 56, 456, 456, 56, 0, 0, 28, 728, 2128, 1128, 84, 0, 0, 8, 728, 5328, 7728, 2472, 120, 0, 0, 1, 486, 8451, 27876, 23607, 4950, 165

Offset: 0

Giuliano Cabrele, Dec 13 2015

In the following description the alphabet {0..r} is taken as a basis, with r = 3 in this case.
For example, the quaternary word 2|03|123|3 of length n=7, has m=4 strictly increasing runs.
The empty word has n = 0 and m = 0, and T(0, 0) = 1.
T(n, 0) = 0 for n >= 1.
T(n, m) <> 0 for m <= n <= m*(r+1). T(m*(r+1), m) = 1.
T(n,m) is a partition, based on m, of all the words of length n, so Sum_{k=0..n} T(n,k) = (r+1)^n.

			Triangle starts:
1;
0, 4;
0, 6, 10;
0, 4, 40,  20;
0, 1, 65, 155,  35;
0, 0, 56, 456, 456, 56;
.
T(3,2) = 40, which accounts for the following words:
[0 <= a <= 0, 1 |    0 <= b <= 1]  =   2
[0 <= a <= 1, 2 |    0 <= b <= 2]  =   6
[0 <= a <= 2, 3 |    0 <= b <= 3]  =  12
[0 <= a <= 3    | 0, 1 <= b <= 3]  =  12
[1 <= a <= 3    | 1, 2 <= b <= 3]  =   6
[2 <= a <= 3    | 2, 3 <= b <= 3]  =   2

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 24, p. 154.

MathPages, Balls In Bins With Limited Capacity

Cf. A119900 (r=1, binary words), A120987 (r=2, ternary words), A008287 (quadrinomial coefficients).
Row sums give A000302.
Cf. A000292.

T:=(n,m)->_plus((-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n)$j=0..m):

T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(4*j, n));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Feb 09 2016

Refer to comment to A120987 concerning formulas for general values of r and considerations.
Therefrom we get
T(n, m) = Qsc(3, n, m) =
Nb(4*m-n, 3, n+1) = Nb(4*(n-m)+3, 3, n+1) =
Sum_{j=0..n+1} (-1)^j*Cb(n+1, j)*Cb(4*(m-j), 4*(m-j)-n) =
Sum_{j=0..m} (-1)^(m-j)*Cb(n+1, m-j)*Cb(4*j, n) =
(in this last version Cb(n,m) can be replaced by binomial(n,m))
Sum_{j=0..m} (-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n) = [z^n, t^m](1-t)/(1-t(1+(1-t)z)^4) where [x^n]F(x) denotes the coefficient of x^n in the formal power series expansion of F(x),
Nb(s,r,n) denotes the (r+1)-nomial coefficient [x^s](1+x+..+x^r)^n,(Nb(s,3,n) = A008287(n,s)).
Cb(x,m) denotes the binomial coefficient in its extended falling factorial notation (Cb(x,m)= x^_m/m! iff m is a nonnegative integer, 0 otherwise), as defined in the Graham et al. reference.
The diagonal T(n, n) = Nb(3, 3, n+1) = Sum_{j=0..n} (-1)^(n-j)*Cb(n+1, n-j)*Cb(4*j, n) = Cb(n+3, 3) = binomial(n+3, 3) = A000292(n+1).

A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 9, 11, 10, 6, 3, 1, 34, 90, 120, 120, 96, 64, 35, 15, 5, 1, 364, 1000, 1400, 1505, 1351, 1044, 700, 406, 202, 84, 28, 7, 1, 4269, 11925, 17225, 19425, 18657, 15753, 11845, 7965, 4785, 2553, 1197, 485, 165, 45, 9, 1

Offset: 0

Michel Marcus, Feb 07 2025

			Triangle begins:
   1
   1    1    1    1
   4    9   11   10    6    3   1
  34   90  120  120   96   64  35  15   5  1
 364 1000 1400 1505 1351 1044 700 406 202 84 28 7 1
 ...

Boualam Rezig and Moussa Ahmia, Combinatorics of three-Catalan numbers and some positivities, arXiv:2502.03615 [math.CO], 2025. See Table 2 p. 5. T(4,7)=405 is a typo.

Row sums are A005721.

Cf. A008287.

t(n, k) = polcoef((1 + x + x^2 + x^3)^n, k); \\ A008287
T(n, k) = t(2*n, 3*n+k) - t(2*n, 3*n+k+1);
row(n) = vector(3*n+1, k, T(n,k-1));

T(n, k) = A008287(2*n, 3*n+k) - A008287(2*n, 3*n+k+1).

A005724 Quadrinomial coefficients.

Original entry on oeis.org

3, 40, 546, 7728, 112035, 1650792, 24608948, 370084832, 5603730876, 85316186400, 1304770191802, 20029132137840, 308437355259930, 4762695514958640, 73716196036213800, 1143325208566357440, 17765127399780725316, 276484586847524844768, 4309270265307160983144

Offset: 1

N. J. A. Sloane

nonn

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).

R. K. Guy, Letter to N. J. A. Sloane, 1987

Cf. A008287.

seq(coeff((1+x+x^2+x^3)^(2*n),x,3*n-1), n=1..30); # Robert Israel, Aug 15 2016

a(n) = A008287(2 * n, 3 * n - 1). - Sean A. Irvine, Aug 15 2016

Offset changed by Robert Israel, Aug 15 2016

A340620 T(n,k) is the number of 4-ary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit with 0 <= k <= n.

Original entry on oeis.org

3, 6, 6, 10, 28, 10, 15, 81, 81, 15, 21, 186, 354, 186, 21, 28, 371, 1137, 1137, 371, 28, 36, 672, 3018, 4836, 3018, 672, 36, 45, 1134, 7023, 16374, 16374, 7023, 1134, 45, 55, 1812, 14829, 47286, 68644, 47286, 14829, 1812, 55, 66, 2772, 29043, 121314, 240021, 240021, 121314, 29043, 2772, 66

Offset: 0

Ji Young Choi, Jan 13 2021

A digit in a string is called indispensable if it is greater than the following digit or equal to the following digits which are eventually greater than the following digit. We also assume that there is an invisible digit 0 at the end of any string. For example, in the string 33102232, the digits 3, 3, 1, 3, and 2 are indispensable (from the left).

T(n,k) is also the number of integers m where the length of base-4 representation of m is n+k and the digit sum of the base-4 representation of 3m is 3(k+1).

			Triangle begins
   3;
   6,   6;
  10,  28,   10;
  15,  81,   81,   15;
  21, 186,  354,  186,   21;
  28, 371, 1137, 1137,  371,  28;
  36, 672, 3018, 4836, 3018, 672, 36;
  ...
There are 6 4-ary strings (10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digits and a nonzero leading digit.
There are 6 4-ary strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits and a nonzero leading digit.
There are 10 4-ary strings (111, 211, 221, 222, 311, 321, 322, 331, 332, 333) of length 3 with 3 indispensable digits and a nonzero leading digit.
Hence, T(1,0)=6, T(1,1)=6, T(2,2)=10.

J. Y. Choi, Indispensable digits for digit sums, Notes on Number Theory and Discrete Mathematics, 25(2), (2019), pp. 40-48.
J. Y. Choi, Digit sums generalizing binomial coefficients, J. Integer Seq. 22 (2019), Article 19.8.3.

Cf. A330381, A330509, A330510, A005773, A008287.

A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
A330509(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);
T(n, k) = A330509(n+1,k+1) - A330509(n,k+1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jan 19 2021

T(n,k) = A330509(n+1,k+1) - A330509(n,k+1).

More terms from Michel Marcus, Jan 19 2021

cp's OEIS Frontend

A330509 Triangle read by rows: T(n,k) is the number of 4-ary strings of length n with k indispensable digits, with 0 <= k <= n.

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A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

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A177960 Numbers of the form A001317(t), excluding those at places of the form t=m*(2^k-1), m>=0, k>=2.

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A178619 Triangle T(n,k) with the coefficient of [x^k] of the series (1-x)^(n+1)* sum_{j>=0} binomial(n + 4*j, 4*j)*x^j in row n, column k.

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A265644 Triangle read by rows: T(n,m) is the number of quaternary words of length n with m strictly increasing runs (0 <= m <= n).

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A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

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A005724 Quadrinomial coefficients.

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A340620 T(n,k) is the number of 4-ary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit with 0 <= k <= n.

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A178619 Triangle T(n,k) with the coefficient of [x^k] of the series (1-x)^(n+1)* sum_{j>=0} binomial(n + 4j, 4j)*x^j in row n, column k.