A136572 → basis A013610 → D
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 3 | 3 | ||||||
| 2 | 12 | 36 | 48 | |||||
| 3 | 54 | 324 | 486 | 864 | ||||
| 4 | 288 | 2592 | 7776 | 7776 | 18432 | |||
| 5 | 1800 | 21600 | 97200 | 194400 | 145800 | 460800 | ||
| 6 | 12960 | 194400 | 1166400 | 3499200 | 5248800 | 3149280 | 13271040 | |
| 7 | 105840 | 1905120 | 14288400 | 57153600 | 128595600 | 154314720 | 77157360 | 433520640 |
Compute more
?
Technical details
- Node id
n-5f838e3b3809278a· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-f9762973ec1089d7
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
- 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: 3; 2: 12, 36; 3: 54, 324, 486; 4: 288, 2592, 7776, 7776; 5: 1800, 21600, 97200, 194400, 145800; 6: 12960, 194400, 1166400, 3499200, 5248800, 3149280; 7: 105840, 1905120, 14288400, 57153600, 128595600, 154314720, 77157360; 8: 967680, 20321280, 182891520, 914457600, 2743372800, 4938071040, 4938071040, 2116316160; 9: 9797760, 235146240, 2469035520, 14814213120, 55553299200, 133327918080, 199991877120, 171421608960, 64283103360;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (n^2+3*n^2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n^2*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n^2*x)*v[i - 1])/(n - 1)); 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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: row n consists of n zeros followed by n!.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle read by rows: row n consists of n zeros followed by n!.?
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
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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Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Geometric basis map (elementary) (above)
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
- ↑ Geometric basis map (elementary) (above)
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
-
-
from
basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
-
-
from
basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.?
Related ?
Same fingerprint ?
- A131431 → basis A013610
- basis-map coefficients of A112227 → θ+1 → basis A013610
- A013610 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013610 → reverse
- A156992 → basis A007318 → x ↦ x+1
- basis-map coefficients of A026729, shifted → D → basis A013610
- basis-map coefficients of A026729 → (θ+1)² → basis A013610
- A027465 → D
- A038243 → basis A013609
- A120909 → basis A013610 → x ↦ x/(x+1)
- basis-map coefficients of A112227 → basis A013610 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013610 → D
- A038330 → basis A013609 → x ↦ x+1
- A136572 → θ+1 → basis A013610
- basis-map coefficients of A030528, shifted → D → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
- A130123 → (θ+1)² → basis A013610
- A178822 → basis A013611 → x ↦ x/(x+1)
- A130123 → θ+1 → basis A013610
- A178821 → basis A007318 → x ↦ x+1
- basis-map coefficients of A030528, shifted → θ+1 → basis A013610
- A085880 → basis A013611 → x ↦ x/(x+1)
- A136572 → D → basis A013610
- A085880 → x ↦ x+1 → x ↦ x+1
- A038325 → basis A007318 → basis A007318
- A013610 → basis A013611 → D
- A028326 → basis A007318 → basis A007318
- A027465 → D → D
- A178820 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013610
- A038330 → D
- A013610 → D → basis A013611
- A038279 → basis A013610 → D
- A127648 → D → basis A013610
- A196347 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A021012 → θ+1 → basis A013610
- A196347 → x ↦ x+1 → basis A007318
- A136572 → (θ+1)² → basis A013610
- A356546 → x ↦ x+1 → x ↦ x+1
- A156992 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013610 → D
- A147716 → basis A013610 → basis A013609
- image under θ+1 → basis A013610
- A132440 → θ+1 → basis A013610
- A120909 → x ↦ x/(x+1)
- basis-map coefficients of A026729 → basis A013610 → reverse
- basis-map coefficients of A026729 → basis A013610
- A038338 → x ↦ x+1 → basis A007318
- A120910 → D → x ↦ x+1
- A085881 → basis A013611 → x ↦ x/(x+1)
- A178822 → x ↦ x+1 → x ↦ x+1
- basis-map coefficients of A112227 → basis A013610 → D
- A013610: Triangle of coefficients in expansion of (1+3*x)^n.
- basis-map coefficients of A112227 → basis A013610
- A013610 → D → D
- A129186 → basis A013610
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
- basis-map coefficients of A021012 → basis A013610
- A028326 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729 → basis A013610 → basis A013611
- A038330 → reverse
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
- A130123 → basis A013610 → D
- basis-map coefficients of A026729 → D → basis A013610
- A013610 → basis A013611 → reverse
- A120910 → basis A007318
- A178820 → basis A007318 → basis A007318
- A038243 → basis A013609 → D
- A038279 → basis A013610 → reverse
- A038221 → basis A013611 → x ↦ x/(x+1)
- A135065 → x ↦ x+1 → basis A007318
- basis-map coefficients of A112227 → (θ+1)² → basis A013610
- A356546 → basis A013611 → x ↦ x/(x+1)
- A038279 → basis A013610
- A127648 → basis A013610
- A178821 → basis A013611 → x ↦ x/(x+1)
- A130123 → basis A013610 → basis A013611
- A136572 → basis A013610 → basis A013611
- A131431 → D → basis A013610
- basis-map coefficients of A112227 → basis A013610 → reverse
- image under θ+1 → θ+1 → D → basis A013610
- A120910 → basis A007318 → reverse
- A131436 → basis A013610
- A027465: Cube of lower triangular normalized binomial matrix.
- A134309 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610
- A130123 → basis A013610
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
- basis-map coefficients of A026729, shifted → θ+1 → basis A013610
- A131431 → θ+1 → basis A013610
- A094305 → x ↦ x+1 → x ↦ x+1
- A085881 → basis A007318 → basis A007318
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
- A136572 → basis A013610
- A136521 → basis A013610
- A130123 → D → basis A013610
- A013609 → basis A013610 → x ↦ x/(x+1)
- image under θ+1 → (θ+1)² → basis A013610
- A038329 → x ↦ x+1 → basis A013609
- A013610 → D
- basis-map coefficients of A026729 → basis A013610 → D
- basis-map coefficients of A026729, shifted → basis A013610 → reverse
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.