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basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
1 · 0, 2 · -2, 2, 3 · 0, -8, 8, 4 …
row sums 1, 2, 3, 4, 5, 6, 7, 8, …
(−∞, 1]
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basis-map coefficients of A145596 → D+1
1 · 2, 2 · 0, 8, 3 · -8, 8, 20, 4 …
row sums 1, 4, 11, 24, 45, 76, 119, 176, …
(−∞, ∞)
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narayana → D+1
1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 …
row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
(−∞, 0]
nonnegative
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A145597 → basis A013609
1 · 6, 6 · 27, 54, 24 · 110, 330, 300, 80 …
row sums 1, 12, 105, 820, 6081, 43932, 313012, …
(−∞, 0]
nonnegative
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A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 …
row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
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A001263 → D+1
1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 …
row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
(−∞, 0]
nonnegative
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A001263 → basis A013610
1 · 2, 3 · 5, 15, 9 · 14, 63, 81, 27 …
row sums 1, 5, 29, 185, 1257, 8925, 65445, …
(−∞, 0]
nonnegative
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A145597 → θ+1
1 · 3, 6 · 6, 30, 18 · 10, 90, 135, 40 …
row sums 1, 9, 54, 275, 1287, 5733, 24752, …
(−∞, 0]
nonnegative
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A145597 → basis A008277
1 · 6, 3 · 27, 33, 6 · 110, 250, 105, 10 …
row sums 1, 9, 66, 475, 3525, 27384, 223538, …
(−∞, 0]
nonnegative
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A001263 → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 …
row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
(−∞, 0]
nonnegative
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A145597 → x ↦ x/(x+1)
1 · 3, 6 · 6, 27, 27 · 10, 75, 165, 110 …
row sums 1, 9, 60, 360, 2061, 11529, 63728, …
[−1, 0]
nonnegative
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A145597 → (θ+1)²
1 · 3, 12 · 6, 60, 54 · 10, 180, 405, 160 …
row sums 1, 15, 120, 755, 4191, 21567, 105560, …
(−∞, 0]
nonnegative
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narayana → D
1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 …
row sums 0, 1, 5, 21, 84, 330, 1287, 5005, …
(−∞, 0]
nonnegative
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narayana
1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 …
row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
(−∞, 0)
nonnegative
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A001263 → basis A013611
1 · 2, 4 · 5, 20, 16 · 14, 84, 144, 64 …
row sums 1, 6, 41, 306, 2426, 20076, 171481, …
(−∞, 0]
nonnegative
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narayana → basis A013611
1 · 2, 4 · 5, 20, 16 · 14, 84, 144, 64 …
row sums 1, 6, 41, 306, 2426, 20076, 171481, …
(−∞, 0]
nonnegative
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A145596 → x ↦ x+1
1 · 4, 2 · 14, 14, 3 · 48, 72, 32, 4 …
row sums 1, 6, 31, 156, 785, 3978, 20335, …
(−∞, −1]
nonnegative
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A145597 → basis A007318
1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 …
row sums 1, 9, 60, 360, 2061, 11529, 63728, …
(−∞, −1]
nonnegative
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A001263 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 …
row sums 1, 5, 22, 95, 406, 1722, 7260, 30459, …
(−∞, 0]
nonnegative
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A145596 → Pₙ + Pₙ₊₁
3, 2 · 5, 10, 3 · 7, 28, 23, 4 · 9, 60, 95, 44, 5 …
row sums 5, 18, 62, 213, 737, 2574, 9074, 32266, …
(−∞, ∞)
nonnegative
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A145597 → basis A013611
1 · 6, 12 · 27, 108, 96 · 110, 660, 1200, 640 …
row sums 1, 18, 231, 2610, 27765, 286146, …
(−∞, 0]
nonnegative
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narayana → x ↦ x/(x+1)
1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 …
row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
[−1, 0]
nonnegative
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A145596 → θ+1
1 · 2, 4 · 3, 16, 9 · 4, 40, 60, 16 …
row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
(−∞, 0]
nonnegative
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A001263 → basis A013609
1 · 2, 2 · 5, 10, 4 · 14, 42, 36, 8 …
row sums 1, 4, 19, 100, 562, 3304, 20071, …
(−∞, 0]
nonnegative
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A145596 → D+1
1 · 4, 2 · 11, 14, 3 · 24, 60, 32, 4 …
row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
(−∞, 0]
nonnegative
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basis-map coefficients of A145596 → D
2 · 2, 6 · -8, 16, 12 · -10, -30, 60, 20 …
row sums 0, 2, 8, 20, 40, 70, 112, 168, …
(−∞, ∞)
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narayana → γ
1 · 1 · 1, 1 · 1, 3 …
row sums 1, 1, 2, 4, 9, 21, 51, 127, …
(−∞, 0]
nonnegative
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narayana → basis A007318
1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 …
row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
(−∞, −1]
nonnegative
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A001263 → basis A008277
1 · 2, 1 · 5, 6, 1 · 14, 31, 12, 1 …
row sums 1, 3, 12, 58, 323, 2014, 13793, 102351, …
(−∞, 0]
nonnegative
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A145596 → D
2 · 8, 6 · 20, 40, 12 · 40, 150, 120, 20 …
row sums 0, 2, 14, 72, 330, 1430, 6006, 24752, …
(−∞, 0]
nonnegative
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A145597 → γ
1 · 3 · 6, 3 · 10, 15 …
row sums 1, 3, 9, 25, 69, 189, 518, 1422, …
(−∞, 0]
nonnegative
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narayana → basis A013610
1 · 2, 3 · 5, 15, 9 · 14, 63, 81, 27 …
row sums 1, 5, 29, 185, 1257, 8925, 65445, …
(−∞, 0]
nonnegative
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A001263 → D
1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 …
row sums 0, 1, 5, 21, 84, 330, 1287, 5005, …
(−∞, 0]
nonnegative
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narayana → basis A008277
1 · 2, 1 · 5, 6, 1 · 14, 31, 12, 1 …
row sums 1, 3, 12, 58, 323, 2014, 13793, 102351, …
(−∞, 0]
nonnegative
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narayana → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 …
row sums 1, 5, 22, 95, 406, 1722, 7260, 30459, …
(−∞, 0]
nonnegative
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narayana → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 …
row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
(−∞, 0]
nonnegative
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A145597 → basis A013610
1 · 6, 9 · 27, 81, 54 · 110, 495, 675, 270 …
row sums 1, 15, 162, 1550, 14019, 123165, …
(−∞, 0]
nonnegative
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A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 …
row sums 1, 6, 27, 110, 429, 1638, 6188, 23256, …
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A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 …
row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
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A145596 → x ↦ x/(x+1)
1 · 2, 4 · 3, 14, 14 · 4, 32, 72, 48 …
row sums 1, 6, 31, 156, 785, 3978, 20335, …
[−1, 0]
nonnegative
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A145596 → (θ+1)²
1 · 2, 8 · 3, 32, 27 · 4, 80, 180, 64 …
row sums 1, 10, 62, 328, 1605, 7502, 34034, …
(−∞, 0]
nonnegative
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A145597 → D+1
1 · 6, 3 · 21, 27, 6 · 55, 135, 75, 10 …
row sums 1, 9, 54, 275, 1287, 5733, 24752, …
(−∞, 0]
nonnegative
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A145597 → D
3 · 15, 12 · 45, 90, 30 · 105, 378, 315, 60 …
row sums 0, 3, 27, 165, 858, 4095, 18564, 81396, …
(−∞, 0]
nonnegative
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narayana → basis A013609
1 · 2, 2 · 5, 10, 4 · 14, 42, 36, 8 …
row sums 1, 4, 19, 100, 562, 3304, 20071, …
(−∞, 0]
nonnegative