A323220 -id:A323220 - cp's OEIS frontend

A323224 A(n, k) = [x^k] C^nx/(1 - x) where C = 2/(1 + sqrt(1 - 4x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 9, 1, 0, 1, 5, 13, 22, 23, 1, 0, 1, 6, 19, 41, 64, 65, 1, 0, 1, 7, 26, 67, 131, 196, 197, 1, 0, 1, 8, 34, 101, 232, 428, 625, 626, 1, 0, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1

Offset: 0

Peter Luschny, Jan 24 2019

Equals A096465 when the leading column (k = 0) is removed. - Georg Fischer, Jul 26 2023

			The square array starts:
   [n\k]  0  1   2   3    4     5     6      7       8       9
    ---------------------------------------------------------------
    [0]   0, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A057427
    [1]   0, 1,  2,  4,   9,   23,   65,   197,    626,   2056, ... A014137
    [2]   0, 1,  3,  8,  22,   64,  196,   625,   2055,   6917, ... A014138
    [3]   0, 1,  4, 13,  41,  131,  428,  1429,   4861,  16795, ... A001453
    [4]   0, 1,  5, 19,  67,  232,  804,  2806,   9878,  35072, ... A114277
    [5]   0, 1,  6, 26, 101,  376, 1377,  5017,  18277,  66727, ... A143955
    [6]   0, 1,  7, 34, 144,  573, 2211,  8399,  31655, 118865, ...
    [7]   0, 1,  8, 43, 197,  834, 3382, 13378,  52138, 201364, ...
    [8]   0, 1,  9, 53, 261, 1171, 4979, 20483,  82499, 327656, ...
    [9]   0, 1, 10, 64, 337, 1597, 7105, 30361, 126292, 515659, ...
.
Triangle given by ascending antidiagonals:
    0;
    0, 1;
    0, 1, 1;
    0, 1, 2,  1;
    0, 1, 3,  4,   1;
    0, 1, 4,  8,   9,   1;
    0, 1, 5, 13,  22,  23,   1;
    0, 1, 6, 19,  41,  64,  65,   1;
    0, 1, 7, 26,  67, 131, 196, 197,   1;
    0, 1, 8, 34, 101, 232, 428, 625, 626, 1;
.
The difference table of a column successively gives the preceding columns, here starting with column 6.
col(6) = 1, 65, 196, 428, 804, 1377, 2211, 3382, 4979, 7105, ...
col(5) =    64, 131, 232, 376,  573,  834, 1171, 1597, 2126, ...
col(4) =         67, 101, 144,  197,  261,  337,  426,  529, ...
col(3) =              34,  43,   53,   64,   76,   89,  103, ...
col(2) =                    9,   10,   11,   12,   13,   14, ...
col(1) =                          1,    1,    1,    1,    1, ...
col(0) =                                0,    0,    0,    0, ...
.
Example for the sum formula: C(0) = 1, C(1) = 1, C(2) = 2 and C(3) = 5.
X(3, 4) = {{0,0,0}, {0,0,1}, {0,1,0}, {1,0,0}, {0,0,2}, {0,1,1}, {0,2,0}, {1,0,1},
{1,1,0}, {2,0,0}, {0,0,3}, {0,1,2}, {0,2,1}, {0,3,0}, {1,0,2}, {1,1,1}, {1,2,0},
{2,0,1}, {2,1,0}, {3,0,0}}. T(3,4) = 1+1+1+1+2+1+2+1+1+2+5+2+2+5+2+1+2+2+2+5 = 41.

The coefficients of the polynomials generating the columns are in A323233.
Sums of antidiagonals and row 1 are A014137. Main diagonal is A242798.
Rows: A057427 (n=0), A014137 (n=1), A014138 (n=2), A001453 (n=3), A114277 (n=4), A143955 (n=5).
Columns: A000027 (k=2), A034856 (k=3), A323221 (k=4), A323220 (k=5).
Similar array based on central binomials is A323222.
Cf. A096465.

Row := proc(n, len) local C, ogf, ser; C := (1-sqrt(1-4*x))/(2*x);
ogf := C^n*x/(1-x); ser := series(ogf, x, (n+1)*len+1);
seq(coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do Row(n, 9) od;
# Alternatively by recurrence:
B := proc(n, k) option remember; if n <= 0 or k < 0 then 0
elif n = k then 1 else B(n-1, k) + B(n, k-1) fi end:
A := (n, k) -> B(n + k, k): seq(lprint(seq(A(n, k), k=0..9)), n=0..9);

(* Illustrating the sum formula, not efficient. *) T[0, K_] := Boole[K != 0];
T[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
Sum[Product[CatalanNumber[m[[i]]], {i, 1, N}], {m , X[K]}]];
Trow[n_] := Table[T[n, k], {k, 0, 9}]; Table[Trow[n], {n, 0, 9}]

For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let C(m) denote the m-th Catalan number. Then: A(n, k) = Sum_{(j1,...,jn) in X(n, k)} C(j1)*C(j2)*...*C(jn).

A(n, k) = T(n + k, k) with T(n, k) = T(n-1, k) + T(n, k-1) with T(n, k) = 0 if n <= 0 or k < 0 and T(n, n) = 1.

A323233 Coefficients of polynomials p(n, x) generating the columns of A323224, triangle read by rows, T(n, k) for n >= 1 and k >= 0.

Original entry on oeis.org

1, 2, 2, 6, 15, 3, 24, 140, 48, 4, 120, 1750, 775, 110, 5, 720, 28644, 14550, 2670, 210, 6, 5040, 588588, 323008, 68775, 7105, 357, 7, 40320, 14592864, 8388800, 1962632, 239120, 16016, 560, 8, 362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9

Offset: 1

Peter Luschny, Jan 27 2019

			The triangle starts:
[ 1]       1;
[ 2]       2,         2;
[ 3]       6,        15,         3;
[ 4]      24,       140,        48,        4;
[ 5]     120,      1750,       775,      110,       5;
[ 6]     720,     28644,     14550,     2670,     210,      6;
[ 7]    5040,    588588,    323008,    68775,    7105,    357,     7;
[ 8]   40320,  14592864,   8388800,  1962632,  239120,  16016,   560,   8;
[ 9]  362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9;
The first few polynomials are:
p[1](x) = 1;
p[2](x) = 2*x + 2!;
p[3](x) = 3*x*(x + 5) + 3!;
p[4](x) = 4*x*(x + 5)*(x + 7) + 4!;
p[5](x) = 5*x*(x + 5)*(x + 7)*(x + 10) + 5!;
p[6](x) = 6*x*(x + 7)*(x + 11)*(x^2 + 17*x + 62) + 6!;
p[7](x) = 7*x*(x + 6)*(x + 7)*(x + 11)*(x + 13)*(x + 14) + 7!;

Cf. A323224, A034856, A323221, A323220, A014137, A014138.

ogf[n_] := (2/(1 + Sqrt[1 - 4 x] ))^n  x/(1 - x);
ser[n_, len_] := CoefficientList[Series[ogf[n], {x, 0, (n + 1) len + 1}], x];
tab[k_, len_] := Table[{n, ser[n, k + 1][[k + 1]]}, {n, 0, len - 1}];
pol[n_] := n! InterpolatingPolynomial[tab[n, n + 1], x] // Expand;
row[n_] := CoefficientList[pol[n], x]; Table[row[n], {n, 1, 9}]

A323224(n, k) = p(k, n)/k!.
T(n, k) = [x^k] p(n, x).
p(n, 1)/n! and p(n, -1)/n! are versions of the partial sums of the Catalan numbers.

cp's OEIS Frontend

A323224 A(n, k) = [x^k] C^nx/(1 - x) where C = 2/(1 + sqrt(1 - 4x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

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A323233 Coefficients of polynomials p(n, x) generating the columns of A323224, triangle read by rows, T(n, k) for n >= 1 and k >= 0.

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cp's OEIS Frontend

A323224 A(n, k) = [x^k] C^n*x/(1 - x) where C = 2/(1 + sqrt(1 - 4*x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

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Maple

Mathematica

Formula

A323233 Coefficients of polynomials p(n, x) generating the columns of A323224, triangle read by rows, T(n, k) for n >= 1 and k >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A323224 A(n, k) = [x^k] C^nx/(1 - x) where C = 2/(1 + sqrt(1 - 4x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.