A156992 → basis A007318 → θ+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(2 + n x\right)\,P_{n}(x) = \left(4 + 4\,x + x^{2} + 4\,n + 6\,n x + 2\,n x^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 4 | 8 | ||||||
| 2 | 24 | 48 | 18 | 90 | |||||
| 3 | 192 | 576 | 432 | 96 | 1296 | ||||
| 4 | 1920 | 7680 | 8640 | 3840 | 600 | 22680 | |||
| 5 | 23040 | 115200 | 172800 | 115200 | 36000 | 4320 | 466560 | ||
| 6 | 322560 | 1935360 | 3628800 | 3225600 | 1512000 | 362880 | 35280 | 11022480 | |
| 7 | 5160960 | 36126720 | 81285120 | 90316800 | 56448000 | 20321280 | 3951360 | 322560 | 293932800 |
Compute more
?
Technical details
- Node id
n-583419ea7c370e8a· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-e620d4a356abb37c
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: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 4, 4; 2: 24, 48, 18; 3: 192, 576, 432, 96; 4: 1920, 7680, 8640, 3840, 600; 5: 23040, 115200, 172800, 115200, 36000, 4320; 6: 322560, 1935360, 3628800, 3225600, 1512000, 362880, 35280; 7: 5160960, 36126720, 81285120, 90316800, 56448000, 20321280, 3951360, 322560; 8: 92897280, 743178240, 1950842880, 2601123840, 2032128000, 975421440, 284497920, 46448640, 3265920; 9: 1857945600, 16721510400, 50164531200, 78033715200, 73156608000, 43893964800, 17069875200, 4180377600, 587865600, 36288000;
%F FormulaThe row polynomials satisfy (2+n*x)*P(n,x) = (4*n+4+(2*n^2+6*n+4)*x+(n^2+2*n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 + (2*n^2 + 6*n + 4)*x + (n^2 + 2*n + 1)*x^2)*P[n - 1, x])/(2 + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 4 + (2*n^2 + 6*n + 4)*x + (n^2 + 2*n + 1)*x^2)*v[i - 1])/(2 + n*x)); 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
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
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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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
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Geometric basis map (elementary) polytool ? step ›
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from
theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle read by rows: row n consists of n zeros followed by n!.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle read by rows: row n consists of n zeros followed by n!.?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
-
from
theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
-
from
basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
Related ?
Same fingerprint ?
- basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
- A085880 → x ↦ x+1 → θ+1
- A196347 → x ↦ x+1 → θ+1
- A038330 → basis A013609 → θ+1
- A038325 → basis A007318 → θ+1
- A038207 → θ+1
- A038221 → x ↦ x+1 → θ+1
- A027466 → basis A013611 → θ+1
- A178822 → x ↦ x+1 → θ+1
- A356546 → x ↦ x+1 → θ+1
- A178820 → basis A007318 → θ+1
- A038208 → basis A007318 → θ+1
- A120910 → θ+1
- A135065 → x ↦ x+1 → θ+1
- A132440 → D+1 → basis A013609
- A178821 → basis A007318 → θ+1
- A131431 → D+1 → basis A013609
- A003506 → x ↦ x+1 → θ+1
- A028326 → basis A007318 → θ+1
- A120910 → D → θ+1
- A094305 → x ↦ x+1 → θ+1
- A038338 → x ↦ x+1 → θ+1
- A085881 → basis A007318 → θ+1
- image under θ+1 → θ+1 → D+1 → basis A013609
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.