A013610 → D → basis A013609
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 3 | 3 | ||||||
| 2 | 24 | 36 | 60 | |||||
| 3 | 144 | 432 | 324 | 900 | ||||
| 4 | 768 | 3456 | 5184 | 2592 | 12000 | |||
| 5 | 3840 | 23040 | 51840 | 51840 | 19440 | 150000 | ||
| 6 | 18432 | 138240 | 414720 | 622080 | 466560 | 139968 | 1800000 | |
| 7 | 86016 | 774144 | 2903040 | 5806080 | 6531840 | 3919104 | 979776 | 21000000 |
Compute more
?
Technical details
- Node id
n-59c0dcb7b3846aec· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-f06e51c139cfae05
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: 24, 36; 3: 144, 432, 324; 4: 768, 3456, 5184, 2592; 5: 3840, 23040, 51840, 51840, 19440; 6: 18432, 138240, 414720, 622080, 466560, 139968; 7: 86016, 774144, 2903040, 5806080, 6531840, 3919104, 979776; 8: 393216, 4128768, 18579456, 46448640, 69672960, 62705664, 31352832, 6718464; 9: 1769472, 21233664, 111476736, 334430208, 627056640, 752467968, 564350976, 241864704, 45349632;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (4*n+6*n*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[((4*n + 6*n*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] = ((4*n + 6*n*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 Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
-
Geometric basis map (elementary) polytool ? step ›
-
from
derivative of Triangle of coefficients in expansion of (1+3*x)^n.?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+3*x)^n.?
-
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
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Cube of lower triangular normalized binomial matrix.?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+3*x)^n.?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
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
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Gamma criterion (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1-x,x/(1-x)).?
-
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
-
from
Riordan array (1-x,x/(1-x)).?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
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?
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
Geometric basis map (elementary) 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?
-
Affine, reversal and Möbius substitutions (elementary) 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
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
from
Triangle of coefficients in expansion of (1+3*x)^n.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
-
from
Cube of lower triangular normalized binomial matrix.?
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle of coefficients in expansion of (1+3*x)^n.?
-
-
from
derivative of Triangle of coefficients in expansion of (1+3*x)^n.?
Related ?
Same fingerprint ?
- A156992 → basis A013609 → basis A013610
- A038221 → basis A013611 → x ↦ x+1
- A216973 → basis A013610
- A028326 → basis A013610
- A038208 → basis A013610 → D
- basis-map coefficients of A113214, shifted → D → basis A013610
- A132200 → basis A013610
- A356546 → basis A013610
- A028326 → basis A013611 → x ↦ x+1
- A135065 → basis A013610
- basis-map coefficients of A127647 → basis A013610
- A038328 → basis A013609 → basis A013610
- A038220 → D
- A038221 → basis A013610
- basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
- A135838 → basis A013610
- A038330 → basis A013611 → basis A013610
- A178820 → basis A013611 → basis A007318
- A135089 → basis A013610
- A120909 → basis A013610 → x ↦ x+1
- A120910 → basis A013610 → basis A013610
- A178820 → basis A013609 → basis A013610
- A013610 → basis A013611 → basis A013609
- A038338 → basis A013609 → basis A013610
- A136572 → basis A013609 → x ↦ x+1
- basis-map coefficients of A026729 → basis A007318 → basis A013610
- A178822 → x ↦ x+1 → basis A013609
- A028326 → basis A013609 → basis A013610
- A085880 → basis A013610 → D
- A038221 → basis A013609 → basis A013610
- A013609 → basis A013610 → x ↦ x+1
- A178821 → basis A013610
- A085881 → basis A007318 → basis A013609
- A085880 → basis A013609 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
- A178821 → basis A013611 → basis A007318
- A085880 → basis A013610
- basis-map coefficients of A112227 → basis A013610 → basis A013609
- A124860 → basis A013610
- A038243 → basis A013611 → D
- A120910 → x ↦ x/(x+1)
- A038338 → basis A013610
- A038334 → D
- A038330 → basis A013609 → x ↦ x/(x+1)
- A196347 → basis A013611 → x ↦ x+1
- A028326 → basis A007318 → basis A013609
- A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
- A156992 → basis A007318 → basis A013609
- A038338 → basis A013610 → reverse
- A094305 → x ↦ x+1 → basis A013609
- A013620: Triangle of coefficients in expansion of (2+3x)^n.
- A120910 → D → basis A013609
- A013620 → D
- A178822 → basis A013610
- A178822 → basis A013609 → basis A013610
- A003506 → D → basis A013610
- A196347 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
- A120910 → basis A013609 → reverse
- A027465 → basis A013611 → basis A013610
- A120910 → basis A013609 → D
- A094305 → x ↦ x+1 → x ↦ x/(x+1)
- A085881 → basis A013609 → basis A013610
- A134059 → basis A013610
- A085881 → basis A013611 → basis A007318
- A027466 → basis A013611 → x ↦ x/(x+1)
- A038291 → basis A013609 → basis A013611
- basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
- A038325 → basis A007318 → basis A013609
- A038334 → reverse
- A156992 → basis A013611 → x ↦ x+1
- A178821 → basis A013609 → basis A013610
- A156992 → basis A013610 → D
- basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
- A094305 → basis A013610
- A344565 → basis A013610
- A131431 → x ↦ x+1 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → basis A013609
- A356546 → basis A013611 → basis A007318
- A038243 → basis A013611
- A196347 → basis A013610 → D
- A178821 → basis A013610 → D
- A038221 → basis A013610 → D
- A178820 → basis A007318 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
- A135065 → x ↦ x+1 → basis A013609
- A038325 → basis A013610
- A085881 → basis A013610
- A356546 → x ↦ x+1 → basis A013609
- A028326 → D → basis A013610
- A028326 → basis A013610 → D
- A085880 → basis A013610 → reverse
- A038325 → basis A013609 → basis A013610
- A038338 → basis A013610 → D
- A038208 → D → basis A013610
- A258758 → basis A013610
- A178820 → basis A013610
- A156992 → basis A013610
- A120910 → basis A013609
- A134058 → basis A013610
- A356546 → basis A013609 → basis A013610
- A178820 → basis A013610 → D
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
- A085880 → D → basis A013610
- basis-map coefficients of A113214, shifted → basis A013610
- A178822 → basis A013611 → basis A007318
- A003506 → x ↦ x+1 → basis A013609
- A178821 → basis A013610 → reverse
- A136572 → basis A013610 → basis A013609
- A135065 → x ↦ x+1 → x ↦ x/(x+1)
- A038330 → basis A013609 → basis A013609
- A085880 → basis A013611 → x ↦ x+1
- A038208 → basis A013610
- A134346 → basis A013610
- A178822 → basis A013610 → D
- A376827 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
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.