A156992 → basis A013611 → θ+1
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 2\,n + 12\,n x + 16\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 4 | 16 | 20 | |||||||||||||
| 2 | 24 | 192 | 288 | 504 | ||||||||||||
| 3 | 192 | 2304 | 6912 | 6144 | 15552 | |||||||||||
| 4 | 1920 | 30720 | 138240 | 245760 | 153600 | 570240 | ||||||||||
| 5 | 23040 | 460800 | 2764800 | 7372800 | 9216000 | 4423680 | 24261120 | |||||||||
| 6 | 322560 | 7741440 | 58060800 | 206438400 | 387072000 | 371589120 | 144506880 | 1175731200 | ||||||||
| 7 | 5160960 | 144506880 | 1300561920 | 5780275200 | 14450688000 | 20808990720 | 16184770560 | 5284823040 | 63959777280 | |||||||
| 8 | 92897280 | 2972712960 | 31213486080 | 166471925760 | 520224768000 | 998831554560 | 1165303480320 | 761014517760 | 214035333120 | 3860160675840 | ||||||
| 9 | 1857945600 | 66886041600 | 802632499200 | 4994157772800 | 18728091648000 | 44947419955200 | 69918208819200 | 68491306598400 | 38526359961600 | 9512681472000 | 255989602713600 | |||||
| 10 | 40874803200 | 1634992128000 | 22072393728000 | 156959244288000 | 686696693760000 | 1977686478028800 | 3845501485056000 | 5022695817216000 | 4237899595776000 | 2092789923840000 | 460413783244800 | 18504391281868800 | ||||
| 11 | 980995276800 | 43163792179200 | 647456882688000 | 5179655061504000 | 25898275307520000 | 87018205033267200 | 203042478410956800 | 331497923936256000 | 372935164428288000 | 276248269946880000 | 121549238776627200 | 24108939922636800 | 1448169752494080000 | |||
| 12 | 25505877196800 | 1224282105446400 | 20200654739865600 | 179561375465472000 | 1010032736993280000 | 3878525710054195200 | 10558208877369753600 | 20685470453622374400 | 29088942825406464000 | 28729820074475520000 | 18961681249153843200 | 7521989255862681600 | 1358136948975206400 | 121993819950101299200 | ||
| 13 | 714164561510400 | 37136557198540800 | 668458029573734400 | 6536034066943180800 | 40850212918394880000 | 176472919807465881600 | 549026861623227187200 | 1254918540853090713600 | 2117675037689590579200 | 2614413626777272320000 | 2300683991563999641600 | 1369002044567008051200 | 494361849426975129600 | 81906105230504755200 | 11006553533275806105600 | |
| 14 | 21424936845312000 | 1199796463337472000 | 23396031035080704000 | 249557664374194176000 | 1715708942572584960000 | 8235402924348407808000 | 28823910235219427328000 | 75295112451185442816000 | 148237252638271340544000 | 219610744649290874880000 | 241571819114219962368000 | 191660286239381127168000 | 103815988379664777216000 | 34400564196811997184000 | 5265392479103877120000 | 1058906357166879277056000 |
Compute more
?
Technical details
- Node id
n-feafafd9a91f4816· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-7cabb77b45806852
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: 4, 16; 2: 24, 192, 288; 3: 192, 2304, 6912, 6144; 4: 1920, 30720, 138240, 245760, 153600; 5: 23040, 460800, 2764800, 7372800, 9216000, 4423680; 6: 322560, 7741440, 58060800, 206438400, 387072000, 371589120, 144506880; 7: 5160960, 144506880, 1300561920, 5780275200, 14450688000, 20808990720, 16184770560, 5284823040; 8: 92897280, 2972712960, 31213486080, 166471925760, 520224768000, 998831554560, 1165303480320, 761014517760, 214035333120; 9: 1857945600, 66886041600, 802632499200, 4994157772800, 18728091648000, 44947419955200, 69918208819200, 68491306598400, 38526359961600, 9512681472000;
%F FormulaThe row polynomials satisfy (1+2*n*x)*P(n,x) = (2*n+2+(4*n^2+12*n+8)*x+(8*n^2+16*n+8)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (4*n^2 + 12*n + 8)*x + (8*n^2 + 16*n + 8)*x^2)*P[n - 1, x])/(1 + 2*n*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] = ((2*n + 2 + (4*n^2 + 12*n + 8)*x + (8*n^2 + 16*n + 8)*x^2)*v[i - 1])/(1 + 2*n*x)); 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 multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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
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!.?
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
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!.?
-
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 ?
- A028326 → basis A013611 → θ+1
- basis-map coefficients of A030528, shifted → basis A013609 → θ+1
- A120909 → basis A013610 → θ+1
- A130123 → basis A013609 → θ+1
- A116666 → x ↦ x/(x+1)
- A038221 → basis A013611 → θ+1
- A085880 → basis A013611 → θ+1
- A178821 → basis A013611 → θ+1
- A136572 → basis A013609 → θ+1
- A013609 → D → θ+1
- A013609 → basis A013610 → θ+1
- basis-map coefficients of A026729 → basis A013609 → θ+1
- A038208 → basis A013611 → θ+1
- A120909 → θ+1
- A116666 → reverse → x ↦ x+1
- A178820 → basis A013611 → θ+1
- A356546 → basis A013611 → θ+1
- A196347 → basis A013611 → θ+1
- A178822 → basis A013611 → θ+1
- A085881 → basis A013611 → θ+1
- A134082 → basis A007318 → x ↦ x+1
- basis-map coefficients of A112227 → basis A013609 → θ+1
- basis-map coefficients of A026729, shifted → basis A013609 → θ+1
- A013609 → θ+1
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.