A003506 → D → θ+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(6 - 5\,n + n^{2}\right)\,P_{n}(x) = \left(n + 2\,n x + n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-6\,n - 6\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 6 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | Pn(1) |
|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||
| 1 | 0 | ||||||
| 2 | 2 | 2 | |||||
| 3 | 6 | 12 | 18 | ||||
| 4 | 12 | 48 | 36 | 96 | |||
| 5 | 20 | 120 | 180 | 80 | 400 | ||
| 6 | 30 | 240 | 540 | 480 | 150 | 1440 | |
| 7 | 42 | 420 | 1260 | 1680 | 1050 | 252 | 4704 |
Compute more
?
Technical details
- Node id
n-124bc02fba57fd6b· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-e99c3824c4864868
Observed on the stored rows ?
- Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
- 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: 0; 1: 0; 2: 2; 3: 6, 12; 4: 12, 48, 36; 5: 20, 120, 180, 80; 6: 30, 240, 540, 480, 150; 7: 42, 420, 1260, 1680, 1050, 252; 8: 56, 672, 2520, 4480, 4200, 2016, 392; 9: 72, 1008, 4536, 10080, 12600, 9072, 3528, 576;
%F FormulaThe row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) + (2*n^2-6*n+(2*n^2-6*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 6 + 12*x.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x] + (2*n^2 - 6*n + (2*n^2 - 6*n)*x)*P[n - 1, 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] = 0; v[2] = 0; v[3] = 2; v[4] = 6 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2] + (2*n^2 - 6*n + (2*n^2 - 6*n)*x)*v[i - 1])/(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
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 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
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
derivative of Exponential Riordan array [x*exp(x),x].?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Exponential Riordan array [x*exp(x),x].?
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Geometric basis map (elementary) polytool ? step ›
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from
Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
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from
Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
binomial?
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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
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array (1-x,x/(1-x)).?
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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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from
Riordan array (1-x,x/(1-x)).?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
binomial?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
-
from
Exponential Riordan array [x*exp(x),x].?
-
Geometric basis map (elementary) polytool ? step ›
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
-
from
derivative of Exponential Riordan array [x*exp(x),x].?
Related ?
Same fingerprint ?
- A156992 → θ+1
- A028326 → θ+1
- A085880 → basis A013609 → D+1
- A038338 → θ+1
- A124860 → D+1
- basis-map coefficients of A026729, shifted → basis A007318 → θ+1
- basis-map coefficients of A113214, shifted → D+1
- A038208 → D+1
- A085880 → D+1
- basis-map coefficients of A113214, shifted → θ+1
- A134346 → D+1
- A038325 → basis A013609 → θ+1
- A085881 → D+1
- A124860 → θ+1
- A178822 → θ+1
- A135065 → D+1
- A135838 → θ+1
- A027465 → basis A013611 → θ+1
- A038221 → θ+1
- A007318 → ×(−(1/2)n+(1/2)n²) → D+1
- basis-map coefficients of A127647 → θ+1
- A028326 → D → D+1
- A134346 → θ+1
- basis-map coefficients of A026729, shifted → basis A007318 → D+1
- A038325 → θ+1
- A085881 → basis A013609 → θ+1
- basis-map coefficients of A127647 → D+1
- A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
- A178820 → basis A013609 → D+1
- A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
- A085880 → θ+1
- A038338 → basis A013609 → θ+1
- A120910 → basis A013610 → D+1
- A027465 → basis A013611 → D+1
- A196347 → D+1
- A131431 → x ↦ x+1 → θ+1
- A344565 → D+1
- basis-map coefficients of A113214, shifted → D → θ+1
- A038338 → D+1
- A093375 → D → D
- A135065 → θ+1
- A028326 → basis A013609 → θ+1
- basis-map coefficients of A026729 → basis A007318 → θ+1
- basis-map coefficients of A113214, shifted → basis A013609 → D+1
- A085880 → D → θ+1
- A356546 → D+1
- A356546 → θ+1
- A196347 → θ+1
- basis-map coefficients of A113214, shifted → D → D+1
- A178822 → basis A013609 → D+1
- A178820 → D+1
- A376827 → θ+1
- A038221 → basis A013609 → D+1
- A094305 → θ+1
- A196347 → basis A013609 → θ+1
- A178822 → basis A013609 → θ+1
- A178820 → θ+1
- A178820 → basis A013609 → θ+1
- A038208 → θ+1
- A038221 → D+1
- A038338 → basis A013609 → D+1
- A093375 → D
- A085880 → basis A013609 → θ+1
- A094305 → D+1
- A376827 → D+1
- A038208 → D → D+1
- A356546 → basis A013609 → θ+1
- A131431 → x ↦ x+1 → D+1
- A028326 → basis A013609 → D+1
- A196347 → basis A013609 → D+1
- A178822 → D+1
- basis-map coefficients of A026729 → basis A007318 → D+1
- A156992 → basis A013609 → D+1
- A038325 → D+1
- basis-map coefficients of A112227 → x ↦ x+1 → θ+1
- A156992 → D+1
- A038221 → basis A013609 → θ+1
- A216973 → θ+1
- A178821 → θ+1
- A156992 → basis A013609 → θ+1
- A178821 → basis A013609 → θ+1
- A038208 → D → θ+1
- A038330 → basis A013611 → D+1
- A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
- A120910 → basis A013610 → θ+1
- A356546 → basis A013609 → D+1
- A178821 → D+1
- A344565 → θ+1
- A028326 → D+1
- A038330 → basis A013611 → θ+1
- basis-map coefficients of A113214, shifted → basis A013609 → θ+1
- A085881 → basis A013609 → D+1
- A085880 → D → D+1
- basis-map coefficients of A112227 → x ↦ x+1 → D+1
- A135838 → D+1
- A085881 → θ+1
- A028326 → D → θ+1
- A131431 → D+1 → basis A007318
- A178821 → basis A013609 → D+1
- A038325 → basis A013609 → D+1
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [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.