A136572 → basis A013610 → basis A013611
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(4\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 4 | 12 | 16 | |||||||||||||
| 2 | 32 | 192 | 288 | 512 | ||||||||||||
| 3 | 384 | 3456 | 10368 | 10368 | 24576 | |||||||||||
| 4 | 6144 | 73728 | 331776 | 663552 | 497664 | 1572864 | ||||||||||
| 5 | 122880 | 1843200 | 11059200 | 33177600 | 49766400 | 29859840 | 125829120 | |||||||||
| 6 | 2949120 | 53084160 | 398131200 | 1592524800 | 3583180800 | 4299816960 | 2149908480 | 12079595520 | ||||||||
| 7 | 82575360 | 1734082560 | 15606743040 | 78033715200 | 234101145600 | 421382062080 | 421382062080 | 180592312320 | 1352914698240 | |||||||
| 8 | 2642411520 | 63417876480 | 665887703040 | 3995326218240 | 14982473318400 | 35957935964160 | 53936903946240 | 46231631953920 | 17336861982720 | 173173081374720 | ||||||
| 9 | 95126814720 | 2568423997440 | 30821087969280 | 215747615784960 | 970864271032320 | 2912592813096960 | 5825185626193920 | 7489524376535040 | 5617143282401280 | 1872381094133760 | 24936923717959680 | |||||
| 10 | 3805072588800 | 114152177664000 | 1541054398464000 | 12328435187712000 | 64724284735488000 | 233007425047756800 | 582518562619392000 | 998603250204672000 | 1123428656480256000 | 748952437653504000 | 224685731296051200 | 3989907794873548800 | ||||
| 11 | 167423193907200 | 5524965398937600 | 82874480984064000 | 745870328856576000 | 4475221973139456000 | 18795932287185715200 | 56387796861557145600 | 120830993274765312000 | 181246489912147968000 | 181246489912147968000 | 108747893947288780800 | 29658516531078758400 | 702223771897744588800 | |||
| 12 | 8036313307545600 | 289307279071641600 | 4773570104682086400 | 47735701046820864000 | 322215982066040832000 | 1546636713916995993600 | 5413228498709485977600 | 13919730425252963942400 | 26099494547349307392000 | 34799326063132409856000 | 31319393456819168870400 | 17083305521901364838400 | 4270826380475341209600 | 134826964204366961049600 | ||
| 13 | 417888291992371200 | 16297643387702476800 | 293357580978644582400 | 3226933390765090406400 | 24202000430738178048000 | 130690802325986161459200 | 522763209303944645836800 | 1568289627911833937510400 | 3528651662801626359398400 | 5881086104669377265664000 | 7057303325603252718796800 | 5774157266402661315379200 | 2887078633201330657689600 | 666248915354153228697600 | 28044008554508327898316800 | |
| 14 | 23401744351572787200 | 982873262766057062400 | 19166028623938112716800 | 229992343487257352601600 | 1897436833769873158963200 | 11384621002619238953779200 | 51230794511786575292006400 | 175648438326125401001164800 | 461077150606079177628057600 | 922154301212158355256115200 | 1383231451818237532884172800 | 1508979765619895490419097600 | 1131734824214921617814323200 | 522339149637656131298918400 | 111929817779497742421196800 | 6281857916209865449222963200 |
Compute more
?
Technical details
- Node id
n-53ee9a2860a05566· revision 0 · JSON- Definition source
derived:basis_map_A013611- Fingerprint
f6-f9762973ec1089d7
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, 12; 2: 32, 192, 288; 3: 384, 3456, 10368, 10368; 4: 6144, 73728, 331776, 663552, 497664; 5: 122880, 1843200, 11059200, 33177600, 49766400, 29859840; 6: 2949120, 53084160, 398131200, 1592524800, 3583180800, 4299816960, 2149908480; 7: 82575360, 1734082560, 15606743040, 78033715200, 234101145600, 421382062080, 421382062080, 180592312320; 8: 2642411520, 63417876480, 665887703040, 3995326218240, 14982473318400, 35957935964160, 53936903946240, 46231631953920, 17336861982720; 9: 95126814720, 2568423997440, 30821087969280, 215747615784960, 970864271032320, 2912592813096960, 5825185626193920, 7489524376535040, 5617143282401280, 1872381094133760;
%F FormulaThe row polynomials satisfy P(n,x) = (4*n+12*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[(4*n + 12*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] = (4*n + 12*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 (−∞, 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_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.?
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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!.?
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A007318 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 ›
-
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 read by rows: row n consists of n zeros followed by n!.?
- ↑ Geometric basis map (elementary) (above)
-
Geometric basis map (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).?
- ↑ 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 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)
- ↑ 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_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
-
-
from
basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.?
Related ?
Same fingerprint ?
- basis-map coefficients of A021012 → θ+1 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → reverse
- A013610 → basis A013611 → D
- A132440 → θ+1 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610 → reverse
- A013610 → basis A013611
- basis-map coefficients of A026729 → (θ+1)² → basis A013610
- A038279 → basis A013610 → reverse
- A136572 → (θ+1)² → basis A013610
- A131436 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013610
- A120909 → basis A013610 → x ↦ x/(x+1)
- basis-map coefficients of A021012 → basis A013610
- basis-map coefficients of A112227 → basis A013610 → basis A013611
- A134309 → basis A013610
- A136572 → θ+1 → basis A013610
- basis-map coefficients of A030528, shifted → θ+1 → basis A013610
- A130123 → D → basis A013610
- basis-map coefficients of A112227 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013610 → basis A013611
- A127648 → basis A013610
- A129186 → basis A013610
- A156992 → basis A007318 → x ↦ x+1
- basis-map coefficients of A030528, shifted → D → basis A013610
- A356546 → basis A013611 → x ↦ x/(x+1)
- A147716 → basis A013610 → basis A013609
- image under θ+1 → (θ+1)² → basis A013610
- A013610 → D
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
- A038330 → basis A013609 → x ↦ x+1
- A038279 → basis A013610
- A013609 → basis A013610 → x ↦ x/(x+1)
- basis-map coefficients of A026729 → basis A013610 → reverse
- A130123 → θ+1 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → D
- A027465 → D
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
- A131431 → D → basis A013610
- basis-map coefficients of A026729 → D → basis A013610
- A038221 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → basis A013610 → D
- A038279 → basis A013610 → D
- A085881 → basis A007318 → basis A007318
- basis-map coefficients of A026729, shifted → basis A013610
- A028326 → basis A013611 → x ↦ x/(x+1)
- A013610 → D → D
- basis-map coefficients of A026729, shifted → D → basis A013610
- A131431 → θ+1 → basis A013610
- A038330 → reverse
- basis-map coefficients of A112227 → basis A013610 → reverse
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
- image under θ+1 → θ+1 → D → basis A013610
- A013610 → D → basis A013611
- A130123 → (θ+1)² → basis A013610
- basis-map coefficients of A112227 → (θ+1)² → basis A013610
- A085880 → x ↦ x+1 → x ↦ x+1
- A178820 → basis A007318 → basis A007318
- A136572 → D → basis A013610
- A356546 → x ↦ x+1 → x ↦ x+1
- A178821 → basis A007318 → x ↦ x+1
- A038330 → D
- A130123 → basis A013610
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
- A178822 → x ↦ x+1 → x ↦ x+1
- A038243 → basis A013609
- basis-map coefficients of A112227 → basis A013610
- A178821 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → θ+1 → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → D
- A196347 → x ↦ x+1 → basis A007318
- basis-map coefficients of A112227 → θ+1 → basis A013610
- A130123 → basis A013610 → basis A013611
- A136572 → basis A013610 → D
- A196347 → basis A013611 → x ↦ x/(x+1)
- A120910 → basis A007318
- A120910 → D → x ↦ x+1
- A094305 → x ↦ x+1 → x ↦ x+1
- A127648 → D → basis A013610
- A135065 → x ↦ x+1 → basis A007318
- A027465: Cube of lower triangular normalized binomial matrix.
- A178822 → basis A013611 → x ↦ x/(x+1)
- A136572 → basis A013610
- A136521 → basis A013610
- A038243 → basis A013609 → D
- A013610: Triangle of coefficients in expansion of (1+3*x)^n.
- A178820 → basis A013611 → x ↦ x/(x+1)
- image under θ+1 → basis A013610
- A120909 → x ↦ x/(x+1)
- A130123 → basis A013610 → D
- A156992 → basis A013611 → x ↦ x/(x+1)
- A085881 → basis A013611 → x ↦ x/(x+1)
- A038338 → x ↦ x+1 → basis A007318
- A027465 → D → D
- basis-map coefficients of A026729 → basis A013610
- A085880 → basis A013611 → x ↦ x/(x+1)
- A028326 → basis A007318 → basis A007318
- A038329 → x ↦ x+1 → basis A013609
- A131431 → basis A013610
- A038325 → basis A007318 → basis A007318
- A120910 → basis A007318 → reverse
- A013610 → basis A013611 → reverse
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
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.