A196347 → basis A013611 → x ↦ x+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 4 | 10 | ||||||
| 2 | 72 | 96 | 32 | 200 | |||||
| 3 | 1296 | 2592 | 1728 | 384 | 6000 | ||||
| 4 | 31104 | 82944 | 82944 | 36864 | 6144 | 240000 | |||
| 5 | 933120 | 3110400 | 4147200 | 2764800 | 921600 | 122880 | 12000000 | ||
| 6 | 33592320 | 134369280 | 223948800 | 199065600 | 99532800 | 26542080 | 2949120 | 720000000 | |
| 7 | 1410877440 | 6584094720 | 13168189440 | 14631321600 | 9754214400 | 3901685760 | 867041280 | 82575360 | 50400000000 |
Compute more
?
Technical details
- Node id
n-06e4f1e201575b5b· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-f06e51c139cfae05
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: 6, 4; 2: 72, 96, 32; 3: 1296, 2592, 1728, 384; 4: 31104, 82944, 82944, 36864, 6144; 5: 933120, 3110400, 4147200, 2764800, 921600, 122880; 6: 33592320, 134369280, 223948800, 199065600, 99532800, 26542080, 2949120; 7: 1410877440, 6584094720, 13168189440, 14631321600, 9754214400, 3901685760, 867041280, 82575360; 8: 67722117120, 361184624640, 842764124160, 1123685498880, 936404582400, 499415777280, 166471925760, 31708938240, 2642411520; 9: 3656994324480, 21941965946880, 58511909191680, 91018525409280, 91018525409280, 60679016939520, 26968451973120, 7705271992320, 1284211998720, 95126814720;
%F FormulaThe row polynomials satisfy P(n,x) = (6*n+4*n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6*n + 4*n*x)*P[n - 1, 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] = (6*n + 4*n*x)*v[i - 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
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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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Geometric basis map (elementary) 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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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)
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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 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).?
-
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).?
-
-
from
basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
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from
basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
Related ?
Same fingerprint ?
- A038220 → D
- A038338 → basis A013610 → reverse
- basis-map coefficients of A026729 → basis A013610 → basis A013609
- A356546 → basis A013609 → basis A013610
- A196347 → basis A013610
- A038208 → D → basis A013610
- basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
- basis-map coefficients of A026729 → basis A007318 → basis A013610
- A085880 → basis A013609 → basis A013610
- A038330 → basis A013611 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
- A038330 → basis A013609 → basis A013609
- A028326 → D → basis A013610
- A131431 → x ↦ x+1 → basis A013610
- A120910 → basis A013609
- A135065 → x ↦ x+1 → x ↦ x/(x+1)
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
- A038328 → basis A013609 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
- A085881 → basis A013609 → basis A013610
- A038208 → basis A013610 → D
- A038243 → basis A013611 → D
- A120910 → x ↦ x/(x+1)
- A156992 → basis A013611 → x ↦ x+1
- A028326 → basis A013609 → basis A013610
- A120910 → basis A013610 → basis A013610
- A085880 → D → basis A013610
- A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
- A038243 → basis A013611
- A038338 → basis A013609 → basis A013610
- A028326 → basis A007318 → basis A013609
- A038208 → basis A013610
- A196347 → basis A013610 → D
- A258758 → basis A013610
- A038334 → reverse
- A038291 → basis A013609 → basis A013611
- A178822 → x ↦ x+1 → basis A013609
- A013609 → basis A013610 → x ↦ x+1
- A178821 → basis A013611 → basis A007318
- A085881 → basis A013611 → basis A007318
- A094305 → basis A013610
- A085880 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
- A013610 → D → basis A013609
- A132200 → basis A013610
- A344565 → basis A013610
- A178820 → basis A013610
- A038221 → basis A013611 → x ↦ x+1
- A156992 → basis A013609 → basis A013610
- A156992 → basis A007318 → basis A013609
- A178822 → basis A013610
- A178821 → basis A013610 → reverse
- A085881 → basis A007318 → basis A013609
- A027465 → basis A013611 → basis A013610
- A136572 → basis A013610 → basis A013609
- A085880 → basis A013611 → x ↦ x+1
- A038338 → basis A013610 → D
- A038334 → D
- A120909 → basis A013610 → x ↦ x+1
- A216973 → basis A013610
- A135065 → basis A013610
- A038325 → basis A007318 → basis A013609
- A135838 → basis A013610
- A085880 → basis A013610 → reverse
- A003506 → D → basis A013610
- basis-map coefficients of A113214, shifted → basis A013610
- A178822 → basis A013610 → D
- A085880 → basis A013610 → D
- A178821 → basis A013610
- A038221 → basis A013610
- A027466 → basis A013611 → x ↦ x/(x+1)
- A120910 → D → basis A013609
- A135089 → basis A013610
- basis-map coefficients of A127647 → basis A013610
- A028326 → basis A013611 → x ↦ x+1
- A124860 → basis A013610
- A038338 → basis A013610
- A028326 → basis A013610
- A178822 → basis A013611 → basis A007318
- A156992 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
- A376827 → basis A013610
- A038325 → basis A013609 → basis A013610
- A013620: Triangle of coefficients in expansion of (2+3x)^n.
- basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
- A136572 → basis A013609 → x ↦ x+1
- A085881 → basis A013610
- A134058 → basis A013610
- A003506 → x ↦ x+1 → basis A013609
- basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
- A094305 → x ↦ x+1 → x ↦ x/(x+1)
- A038325 → basis A013610
- A120910 → basis A013609 → D
- A120910 → basis A013609 → reverse
- A038221 → basis A013609 → basis A013610
- A134059 → basis A013610
- A038330 → basis A013609 → x ↦ x/(x+1)
- A356546 → x ↦ x+1 → basis A013609
- A178821 → basis A013610 → D
- A178820 → basis A007318 → basis A013609
- A028326 → basis A013610 → D
- A356546 → basis A013610
- A094305 → x ↦ x+1 → basis A013609
- A134346 → basis A013610
- basis-map coefficients of A113214, shifted → D → basis A013610
- A178822 → basis A013609 → basis A013610
- A178820 → basis A013609 → basis A013610
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
- A013610 → basis A013611 → basis A013609
- A135065 → x ↦ x+1 → basis A013609
- A156992 → basis A013610
- A013620 → D
- A038221 → basis A013610 → D
- basis-map coefficients of A112227 → basis A013610 → basis A013609
- A178820 → basis A013611 → basis A007318
- A178820 → basis A013610 → D
- A178821 → basis A013609 → basis A013610
- A356546 → basis A013611 → basis A007318
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.