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 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | 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 | |||||||
| 8 | 67722117120 | 361184624640 | 842764124160 | 1123685498880 | 936404582400 | 499415777280 | 166471925760 | 31708938240 | 2642411520 | 4032000000000 | ||||||
| 9 | 3656994324480 | 21941965946880 | 58511909191680 | 91018525409280 | 91018525409280 | 60679016939520 | 26968451973120 | 7705271992320 | 1284211998720 | 95126814720 | 362880000000000 | |||||
| 10 | 219419659468800 | 1462797729792000 | 4388393189376000 | 7801587892224000 | 9101852540928000 | 7281482032742400 | 4045267795968000 | 1541054398464000 | 385263599616000 | 57076088832000 | 3805072588800 | 36288000000000000 | ||||
| 11 | 14481697524940800 | 106199115182899200 | 353997050609664000 | 707994101219328000 | 943992134959104000 | 881059325961830400 | 587372883974553600 | 279701373321216000 | 93233791107072000 | 20718620246016000 | 2762482699468800 | 167423193907200 | 3991680000000000000 | |||
| 12 | 1042682221795737600 | 8341457774365900800 | 30585345172674969600 | 67967433717055488000 | 101951150575583232000 | 108747893947288780800 | 84581695292335718400 | 48332397309906124800 | 20138498879127552000 | 5966962630852608000 | 1193392526170521600 | 144653639535820800 | 8036313307545600 | 479001600000000000000 | ||
| 13 | 81329213300067532800 | 704853181933918617600 | 2819412727735674470400 | 6891897778909426483200 | 11486496298182377472000 | 13783795557818852966400 | 12252262718061202636800 | 8168175145374135091200 | 4084087572687067545600 | 1512625026921136128000 | 403366673845636300800 | 73339395244661145600 | 8148821693851238400 | 417888291992371200 | 62270208000000000000000 | |
| 14 | 6831653917205672755200 | 63762103227252945715200 | 276302447318096098099200 | 736806526181589594931200 | 1350811964666247590707200 | 1801082619554996787609600 | 1801082619554996787609600 | 1372253424422854695321600 | 800481164246665238937600 | 355769406331851217305600 | 118589802110617072435200 | 28749042935907169075200 | 4791507155984528179200 | 491436631383028531200 | 23401744351572787200 | 8717829120000000000000000 |
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).?
-
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 ›
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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!.?
-
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 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).?
-
-
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.