A359363: Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.
A359363 ↗ palindromicγ-positivenonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A359363
\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-1 - x + 2\,n + 5\,n x - n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x) + \left(4 + 8\,x + 4\,x^{2} - 6\,n - 12\,n x - 6\,n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 1 | 1 | ||||||
| 2 | 0 | 1 | 1 | 2 | |||||
| 3 | 0 | 1 | 4 | 1 | 6 | ||||
| 4 | 0 | 1 | 10 | 10 | 1 | 22 | |||
| 5 | 0 | 1 | 20 | 50 | 20 | 1 | 92 | ||
| 6 | 0 | 1 | 35 | 175 | 175 | 35 | 1 | 422 | |
| 7 | 0 | 1 | 56 | 490 | 980 | 490 | 56 | 1 | 2074 |
All 15 rows b-files: A001181 (row sums), A056939, A056939 (reversed triangle), A359363 ?
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Technical details
- Node id
n-afb2f273dc5c03d1· revision 0 · JSON- Definition source
oeis_catalog:A359363- Fingerprint
f6-509399d8a74ba170
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- Palindromic rows: yes
- γ-positive (gamma_positive): yes
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
- A359363 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
- A056939 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
- A056939 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
Row sums ? A001181
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 0, 1; 2: 0, 1, 1; 3: 0, 1, 4, 1; 4: 0, 1, 10, 10, 1; 5: 0, 1, 20, 50, 20, 1; 6: 0, 1, 35, 175, 175, 35, 1; 7: 0, 1, 56, 490, 980, 490, 56, 1; 8: 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1; 9: 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1;
%F FormulaThe row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(n^2-2*n+1-(8*n^2+5*n-1)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) + (2*n^2-6*n+4+(4*n^2-12*n+8)*x+(2*n^2-6*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*P[n - 2, x])/(n^2 + 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*v[i - 2])/(n^2 + 5*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A056939 triangle, offset 0, from P1, low zeros dropped
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 10, 10, 1; 4: 1, 20, 50, 20, 1; 5: 1, 35, 175, 175, 35, 1; 6: 1, 56, 490, 980, 490, 56, 1; 7: 1, 84, 1176, 4116, 4116, 1176, 84, 1; 8: 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1; 9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;
%F FormulaRow n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n^2+7*n+12)*P(n,x) = -(n^2-(8*n^2+21*n+12)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.
%t MathematicaP[0, x_] := x;
P[1, x_] := x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - (8*n^2 + 21*n + 12)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - (8*n^2 + 21*n + 12)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A056939 reversed triangle, offset 0, from P1, low zeros dropped
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 4, 1; 3: 1, 10, 10, 1; 4: 1, 20, 50, 20, 1; 5: 1, 35, 175, 175, 35, 1; 6: 1, 56, 490, 980, 490, 56, 1; 7: 1, 84, 1176, 4116, 4116, 1176, 84, 1; 8: 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1; 9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (n^2+7*n+12)*P(n,x) = -(n^2-(8*n^2+21*n+12)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.
%t MathematicaP[0, x_] := x;
P[1, x_] := x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - (8*n^2 + 21*n + 12)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - (8*n^2 + 21*n + 12)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A359363 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 0, 1; 2: 0, 1, 1; 3: 0, 1, 4, 1; 4: 0, 1, 10, 10, 1; 5: 0, 1, 20, 50, 20, 1; 6: 0, 1, 35, 175, 175, 35, 1; 7: 0, 1, 56, 490, 980, 490, 56, 1; 8: 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1; 9: 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1;
%F FormulaThe row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(n^2-2*n+1-(8*n^2+5*n-1)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) + (2*n^2-6*n+4+(4*n^2-12*n+8)*x+(2*n^2-6*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*P[n - 2, x])/(n^2 + 5*n + 6)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*v[i - 2])/(n^2 + 5*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)
Proof chains ? JSON
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Used in 6
- shift_x_plus_one of Triangle read by rows. The coefficients of the Baxter polyn… Image under a real-rootedness preserver (affine, shift 0)
- reversal of Triangle read by rows. The coefficients of the Baxter polynomials p… Image under a real-rootedness preserver (reversal, shift 0)
- theta_plus_one of Triangle read by rows. The coefficients of the Baxter polynom… Image under a real-rootedness preserver (euler polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows. The coefficients of the Baxt… Image under a real-rootedness preserver (mobius, shift 0)
- derivative of Triangle read by rows. The coefficients of the Baxter polynomials… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- one_plus_derivative of Triangle read by rows. The coefficients of the Baxter po… Image under a real-rootedness preserver (derivative polynomial, shift 0)
Related ?
Same fingerprint ?
- A359363 → reverse
- Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,…
- A056939: Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions wit…
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.