Ji Young Choi - cp's OEIS frontend

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.

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

A330510 Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 4, 10, 4, 5, 22, 22, 5, 6, 40, 70, 40, 6, 7, 65, 171, 171, 65, 7, 8, 98, 356, 534, 356, 98, 8, 9, 140, 665, 1373, 1373, 665, 140, 9, 10, 192, 1148, 3088, 4246, 3088, 1148, 192, 10, 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11

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 ternary representation of m is n+k and the digit sum of the ternary representation of 2m is 2(k+1).

			Triangle begins
  2;
  3,   3;
  4,  10,   4;
  5,  22,  22,   5;
  6,  40,  70,  40,   6;
  7,  65, 171, 171,  65,   7;
  ...
There are 4 strings (100, 112, 120, 200) of length 3 with 1 indispensable digits and a nonzero leading digit.
There are 10 strings (101, 102, 110, 121, 122, 201, 202, 210, 212, 220) of length 3 with 2 indispensable digits are a nonzero leading digit.
There are 4 strings (111, 211, 221, 222) of length 3 with 3 indispensable digits and a nonzero leading digit.
Hence T(2,0)=4, T(2,1)=10, T(2,2)=4.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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. A005773, A330381, A027907.

A027907(n, k) = if(n<0, 0, polcoef((1 + x + x^2)^n, k));
T(n,k) = {A027907(n+1, 2*k+1) + A027907(n+1, 2*k+2) - A027907(n, 2*k+1) - A027907(n, 2*k+2)} \\ Andrew Howroyd, Dec 20 2019

T(n, k) = A330381(n+1, k+1) - A330381(n, k+1).

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.

Original entry on oeis.org

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).

A330381 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 13, 4, 1, 14, 35, 26, 5, 1, 20, 75, 96, 45, 6, 1, 27, 140, 267, 216, 71, 7, 1, 35, 238, 623, 750, 427, 105, 8, 1, 44, 378, 1288, 2123, 1800, 770, 148, 9, 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10, 1, 65, 825, 4302, 11505, 17303

Offset: 0

Ji Young Choi, Dec 12 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 ternary representation of m is less than or equal to n and the digit sum of the ternary representation of 2m is 2k.

			Triangle begins
  1;
  1,  2;
  1,  5,  3;
  1,  9, 13,  4;
  1, 14, 35, 26,  5;
  1, 20, 75, 96, 45, 6;
  ...
There is 1 string (00) of length 2 with 0 indispensable digits.
There are 5 strings (01, 02, 10, 20, 12) of length 2 with 1 indispensable digit.
There are 3 strings (11, 21, 22) of length 2 with 2 indispensable digits.
Hence T(2, 0) = 1, T(2, 1) = 5, T(2, 2) = 3.

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. A027907, A330509.

Table[Total@ Map[Sum[Binomial[n, i] Binomial[n - i, # - 2 i], {i, 0, n}] &, 2 k + {-1, 0}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 23 2019, after Adi Dani at A027907 *)

A027907(n, k) = if(n<0, 0, polcoeff((1 + x + x^2)^n, k));
T(n, k) = A027907(n, 2*k-1) + A027907(n, 2*k); \\ Jinyuan Wang, Dec 14 2019

T(n, k) = A027907(n, 2k-1) + A027907(n, 2k).

More terms from Jinyuan Wang, Dec 14 2019

A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The sequence shows the case of m=3.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, 0, 11, -6, 1, 0, -20, 35, -10, 1, 0, 40, -135, 85, -15, 1, 0, 0, 490, -525, 175, -21, 1, 0, 0, -1120, 2905, -1540, 322, -28, 1, 0, 0, 2240, -12600, 11865, -3780, 546, -36, 1, 0, 0, 0, 47600, -76545, 38325, -8190, 870, -45, 1

Offset: 1

Ji Young Choi, Jan 21 2010

A(n,k,m) is also the (n,k)-th entry in the matrix inverting the matrix consisting of the m-restrained Stirling numbers of the second kind.

Also the Bell transform of the sequence "g(n) = [1,-1,2][n] if n < 3, otherwise 0" (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			A(1,1,3)=1,  A(1,2,3)=0,   A(1,3,3)=0,   A(1,4,3)=0, ...
A(2,1,3)=-1, A(2,2,3)=1,   A(2,3,3)=0,   A(2,4,3)=0, ...
A(3,1,3)=2,  A(3,2,3)=-3,  A(3,3,3)=1,   A(3,4,3)=0, ...
A(4,1,3)=0,  A(4,2,3)=11,  A(4,3,3)=-6,  A(4,4,3)=1, ...

J. Y. Choi, Multi-restrained Stirling numbers

Cf. A111246, A144633, A171998 (matrix inverse).

T(n,k) = (-1)^(n-k)*n!*sum(j=0, k, binomial(j,n-k-j)*binomial(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!); \\ Jinyuan Wang, Dec 21 2019

# uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: [1,-1,2][n] if n < 3 else 0, 12) # Peter Luschny, Jan 19 2016

A(n,k,m) = Sum {(-1)^(n-k)*n!} /{1^{k_1}*2^{k_2}*...*m^{k_m}*(k_1!)*(k_2!)*...*(k_m!)}, where k_1 + 2*k_2 + ... + m*k_m = n and k_1 + k_2 + ... + k_m = k.
Recurrence A(n,k,m) = Sum_{i=1..m} (-1)^(i-1)*[n-1]_{i-1}*A(n-i,k-1,m).
A(n,k,m) = A(n-1,k-1,m) - (n-1)*A(n-1,k,m) - (-1)^m(n-1)*(n-2)*...*(n-m)*A(n-m-1,k-1,m).
Generating function f(t) = (1 + t - t^2/2 + t^3/3 + ... + (-1)^(m-1) t^m/m)^x, for an indeterminate x ===> the n-th derivative of f(t) at t=0, f^(n)(0) = Sum_{k=1..n} A(n,k,m)[x]_k, where [x]_k is the k-th falling factorial
T(n,k) = (-1)^(n-k)*n!*Sum_{j=0..k} C(j,n-k-j)*C(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!. - Vladimir Kruchinin, Oct 02 2019

More terms from Peter Luschny, Jan 19 2016

A171998 In general, let A(n,k,m) denote the (n,k)-th entry of the inverse of the matrix consisting of the (n,k)-th m-restrained Stirling numbers of the second kind (-1)^(n-k) times the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, as the (n+1,k+1)-th entry. The sequence shows A(n,k,3), which is a lower triangular matrix, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, -5, 7, 6, 1, -65, -15, 25, 10, 1, -455, -455, 0, 65, 15, 1, -1295, -4725, -1715, 140, 140, 21, 1

Offset: 1

Ji Young Choi, Jan 21 2010

A(n,k,m) also can be expanded for nonpositive integers n and k using the m-restrained Stirling numbers of the first kind.

			A(1,1,3) = 1, A(1,2,3) = 0, A(1,3,3) = 0, A(1,4,3) = 0, ...
A(2,1,3) = 1, A(2,2,3) = 1, A(2,3,3) = 0, A(2,4,3) = 0, ...
A(3,1,3) = 1, A(3,2,3) = 3, A(3,3,3) = 1, A(3,4,3) = 0, ...
A(4,1,3) = -5, A(4,2,3) = 7, A(4,3,3) = 6, A(4,4,3) = 1, ...
In other words, A(n,k,3) is the matrix
1
1 1
1 3 1
-5 7 6 1
...
with all other entries in each row being 0. - _N. J. A. Sloane_, Dec 21 2019

Ji Young Choi, Multi-restrained Stirling numbers, Ars Comb. 120 (2015), 113-127.
John Engbers, David Galvin, and Cliff Smyth, Restricted Stirling and Lah number matrices and their inverses, Journal of Combinatorial Theory, Series A, 161 (2019), 271-298.

Cf. A111246, A144633.

A(n,k,m) = A(n-1,k-1,m) - Sum_{i=1..m-1} (-1)^{i}(k)...(k+i-1) A(n,k+i,m) A(n,k,m) = A(n-1,k-1,m) + k A(n-1,k,m) + (-1)^m k(k+1)...(k+m-1)A(n,k+m,m).

A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 0, 7, 6, 1, 0, 10, 25, 10, 1, 0, 10, 75, 65, 15, 1, 0, 0, 175, 315, 140, 21, 1, 0, 0, 280, 1225, 980, 266, 28, 1, 0, 0, 280, 3780, 5565, 2520, 462, 36, 1, 0, 0, 0, 9100, 26145, 19425, 5670, 750, 45, 1, 0, 0, 0, 15400, 102025, 125895, 56595, 11550, 1155

Offset: 1

Ji Young Choi, Oct 31 2005

a(n,k) = 0 if k > n; a(n,k) = 0 if n > 0 and k < 0; a(n,k) can be extended to negative n and k, just as the Stirling numbers or Pascal's triangle can be extended. The present triangle is called the tri-restricted Stirling numbers of the second kind.

Also the Bell transform of the sequence "a(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			a(1,1)=1;
a(2,1)=1; a(2,2)=1;
a(3,1)=1; a(3,2)=3; a(3,3)=1;
a(4,1)=0; a(4,2)=7; a(4,3)=6; a(4,4)=1;
a(5,1)=0; a(5,2)=10; a(5,3)=25; a(5,4)=10; a(5,5)=1;
a(6,1)=0; a(6,2)=10; a(6,3)=75; a(6,4)=65; a(6,5)=15; a(6,6)=1; ...

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.
J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

A144385 and A144402 are other versions of this same triangle.

Cf. A001680, A008277 (Stirling numbers).

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0,...) as column 0.
BellMatrix(n -> `if`(n<3,1,0), 10); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# < 3, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

row(n) = {x='x+O('x^(n+1)); polcoeff(serlaplace(exp(y*(x+x^2/2+x^3/6))), n, 'x); }
tabl(nn) = for(n=1, nn, print(Vecrev(row(n)/y))) \\ Jinyuan Wang, Dec 21 2019

a(n, k) = a(n-1, k-1) + k*a(n-1, k) - binomial(n-1, 3)*a(n-4, k-1).
G.f. = Sum_{k_1+k_2+k_3=k, k_1+ 2k_2+3k_3=n} frac{n!}{(1!)^{k_1}(2!)^{k_2}(3!)^{k_3}k_1!k_2!k_3!}.
E.g.f.: exp(y*(x+x^2/2+x^3/6)). - Vladeta Jovovic, Nov 01 2005

More terms from Vladeta Jovovic, Nov 01 2005

Recurrence, offset and example corrected by David Applegate, Jan 16 2009

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User: Ji Young Choi

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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A330510 Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.

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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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A330381 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.

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A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The sequence shows the case of m=3.

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A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

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