A156992 → basis A013611 → basis A013609
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(6 + 8\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 12 | 16 | 28 | ||||||
| 2 | 216 | 576 | 384 | 1176 | |||||
| 3 | 5184 | 20736 | 27648 | 12288 | 65856 | ||||
| 4 | 155520 | 829440 | 1658880 | 1474560 | 491520 | 4609920 | |||
| 5 | 5598720 | 37324800 | 99532800 | 132710400 | 88473600 | 23592960 | 387233280 | ||
| 6 | 235146240 | 1881169920 | 6270566400 | 11147673600 | 11147673600 | 5945425920 | 1321205760 | 37948861440 | |
| 7 | 11287019520 | 105345515520 | 421382062080 | 936404582400 | 1248539443200 | 998831554560 | 443925135360 | 84557168640 | 4250272481280 |
Compute more
?
Technical details
- Node id
n-eda7177349215245· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-c91d5d506c107735
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: 12, 16; 2: 216, 576, 384; 3: 5184, 20736, 27648, 12288; 4: 155520, 829440, 1658880, 1474560, 491520; 5: 5598720, 37324800, 99532800, 132710400, 88473600, 23592960; 6: 235146240, 1881169920, 6270566400, 11147673600, 11147673600, 5945425920, 1321205760; 7: 11287019520, 105345515520, 421382062080, 936404582400, 1248539443200, 998831554560, 443925135360, 84557168640; 8: 609499054080, 6501323243520, 30339508469760, 80905355919360, 134842259865600, 143831743856640, 95887829237760, 36528696852480, 6088116142080; 9: 36569943244800, 438839318937600, 2340476367667200, 7281482032742400, 14562964065484800, 19417285420646400, 17259809262796800, 9862748150169600, 3287582716723200, 487049291366400;
%F FormulaThe row polynomials satisfy P(n,x) = (6*n+6+(8*n+8)*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 + 6 + (8*n + 8)*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 + 6 + (8*n + 8)*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 (−∞, 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 ≥ 1
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) = 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)
-
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!.?
-
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
basis_map_A013609 of 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
basis_map_A013609 of 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.?
-
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.?
-
-
from
basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
-
from
basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
-
from
basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
Related ?
Same fingerprint ?
- A356546 → x ↦ x+1 → basis A013611
- basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
- basis-map coefficients of A112227 → basis A013610 → basis A007318
- A038335 → reverse
- basis-map coefficients of A112227 → basis A013609 → basis A013609
- basis-map coefficients of A026729 → basis A013610 → basis A007318
- A085881 → basis A013611 → basis A013609
- A013610 → basis A007318
- A038334 → basis A013609 → basis A013610
- A120910 → D → basis A013611
- A178822 → basis A013611 → basis A013609
- A038329 → basis A013610 → basis A013609
- A003506 → x ↦ x+1 → basis A013611
- A178822 → x ↦ x+1 → basis A013611
- A120909 → basis A013610 → basis A013609
- A028326 → basis A007318 → basis A013611
- basis-map coefficients of A026729 → basis A013609 → basis A013609
- A356546 → basis A013611 → basis A013609
- A120910 → basis A013611 → reverse
- A196347 → basis A013611 → basis A013609
- A027465 → D → basis A013610
- A038335 → D
- A013609 → D → basis A013609
- A085880 → x ↦ x+1 → basis A013611
- A094305 → x ↦ x+1 → basis A013611
- A013610 → basis A013611 → x ↦ x+1
- A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
- A196347 → x ↦ x+1 → basis A013611
- A038330 → basis A013610 → reverse
- A038329 → basis A013611 → basis A013610
- A028326 → basis A013611 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
- A038243 → basis A013609 → x ↦ x/(x+1)
- A120910 → basis A013611 → D
- A038284 → reverse
- A038284 → D
- Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
- A038330 → basis A013610
- A013610 → basis A007318 → D
- A178820 → basis A013611 → basis A013609
- A130123 → basis A013609 → basis A013609
- A085880 → basis A013611 → basis A013609
- A120910 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
- A178821 → basis A013611 → basis A013609
- A038279 → basis A013610 → x ↦ x/(x+1)
- A038221 → basis A013611 → basis A013609
- A130123 → basis A013610 → x ↦ x+1
- A135065 → x ↦ x+1 → basis A013611
- A038208 → basis A013611 → basis A013609
- A136572 → basis A013610 → basis A007318
- A178821 → basis A007318 → basis A013611
- A038330 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
- A178820 → basis A007318 → basis A013611
- A038330 → basis A013610 → D
- A013609 → basis A013609
- A027465 → x ↦ x/(x+1)
- A013609 → basis A013609 → D
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.