A196347 → basis A013611 → x ↦ x/(x+1)
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 2 | 6 | 8 | |||||||||||||
| 2 | 8 | 48 | 72 | 128 | ||||||||||||
| 3 | 48 | 432 | 1296 | 1296 | 3072 | |||||||||||
| 4 | 384 | 4608 | 20736 | 41472 | 31104 | 98304 | ||||||||||
| 5 | 3840 | 57600 | 345600 | 1036800 | 1555200 | 933120 | 3932160 | |||||||||
| 6 | 46080 | 829440 | 6220800 | 24883200 | 55987200 | 67184640 | 33592320 | 188743680 | ||||||||
| 7 | 645120 | 13547520 | 121927680 | 609638400 | 1828915200 | 3292047360 | 3292047360 | 1410877440 | 10569646080 | |||||||
| 8 | 10321920 | 247726080 | 2601123840 | 15606743040 | 58525286400 | 140460687360 | 210691031040 | 180592312320 | 67722117120 | 676457349120 | ||||||
| 9 | 185794560 | 5016453120 | 60197437440 | 421382062080 | 1896219279360 | 5688657838080 | 11377315676160 | 14627977297920 | 10970982973440 | 3656994324480 | 48704929136640 | |||||
| 10 | 3715891200 | 111476736000 | 1504935936000 | 12039487488000 | 63207309312000 | 227546313523200 | 568865783808000 | 975198486528000 | 1097098297344000 | 731398864896000 | 219419659468800 | 3896394330931200 | ||||
| 11 | 81749606400 | 2697737011200 | 40466055168000 | 364194496512000 | 2185166979072000 | 9177701312102400 | 27533103936307200 | 58999508434944000 | 88499262652416000 | 88499262652416000 | 53099557591449600 | 14481697524940800 | 342882701121945600 | |||
| 12 | 1961990553600 | 70631659929600 | 1165422388838400 | 11654223888384000 | 78666011246592000 | 377596853983641600 | 1321588988942745600 | 3398371685852774400 | 6371946910973952000 | 8495929214631936000 | 7646336293168742400 | 4170728887182950400 | 1042682221795737600 | 32916739307706777600 | ||
| 13 | 51011754393600 | 1989458421350400 | 35810251584307200 | 393912767427379200 | 2954345755705344000 | 15953467080808857600 | 63813868323235430400 | 191441604969706291200 | 430743611181839155200 | 717906018636398592000 | 861487222363678310400 | 704853181933918617600 | 352426590966959308800 | 81329213300067532800 | 3423340888001504870400 | |
| 14 | 1428329123020800 | 59989823166873600 | 1169801551754035200 | 14037618621048422400 | 115810353623649484800 | 694862121741896908800 | 3126879547838536089600 | 10720729878303552307200 | 28141915930546824806400 | 56283831861093649612800 | 84425747791640474419200 | 92100815772698699366400 | 69075611829524024524800 | 31881051613626472857600 | 6831653917205672755200 | 383414179456168545484800 |
Compute more
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Technical details
- Node id
n-816ba7e3126bda81· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- 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: 2, 6; 2: 8, 48, 72; 3: 48, 432, 1296, 1296; 4: 384, 4608, 20736, 41472, 31104; 5: 3840, 57600, 345600, 1036800, 1555200, 933120; 6: 46080, 829440, 6220800, 24883200, 55987200, 67184640, 33592320; 7: 645120, 13547520, 121927680, 609638400, 1828915200, 3292047360, 3292047360, 1410877440; 8: 10321920, 247726080, 2601123840, 15606743040, 58525286400, 140460687360, 210691031040, 180592312320, 67722117120; 9: 185794560, 5016453120, 60197437440, 421382062080, 1896219279360, 5688657838080, 11377315676160, 14627977297920, 10970982973440, 3656994324480;
%F FormulaThe row polynomials satisfy (1+x)*P(n,x) = (2*n+8*n*x+6*n*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 + 8*n*x + 6*n*x^2)*P[n - 1, x])/(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] = ((2*n + 8*n*x + 6*n*x^2)*v[i - 1])/(1 + 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 [-1, 0]
- 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)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (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 ›
-
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!.?
-
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 ?
- A130123 → basis A013610 → basis A013611
- A013609 → basis A013610 → x ↦ x/(x+1)
- A028326 → basis A013611 → x ↦ x/(x+1)
- A038243 → basis A013609
- A027465 → D
- A130123 → θ+1 → basis A013610
- basis-map coefficients of A021012 → θ+1 → basis A013610
- A134309 → basis A013610
- basis-map coefficients of A026729, shifted → θ+1 → basis A013610
- image under θ+1 → basis A013610
- A136572 → θ+1 → basis A013610
- A178821 → basis A007318 → x ↦ x+1
- A178821 → basis A013611 → x ↦ x/(x+1)
- A085881 → basis A007318 → basis A007318
- A136572 → basis A013610
- A136572 → (θ+1)² → basis A013610
- A038279 → basis A013610
- basis-map coefficients of A112227 → (θ+1)² → basis A013610
- A120910 → basis A007318
- A038279 → basis A013610 → D
- A094305 → x ↦ x+1 → x ↦ x+1
- image under θ+1 → θ+1 → D → basis A013610
- A130123 → basis A013610
- A178822 → basis A013611 → x ↦ x/(x+1)
- A013610 → D
- basis-map coefficients of A026729 → (θ+1)² → basis A013610
- A028326 → basis A007318 → basis A007318
- A156992 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A112227 → basis A013610 → basis A013611
- basis-map coefficients of A026729 → D → basis A013610
- A136572 → basis A013610 → D
- basis-map coefficients of A030528, shifted → basis A013610
- A127648 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → reverse
- A038330 → basis A013609 → x ↦ x+1
- A038329 → x ↦ x+1 → basis A013609
- A120910 → D → x ↦ x+1
- A132440 → θ+1 → basis A013610
- A127648 → D → basis A013610
- A135065 → x ↦ x+1 → basis A007318
- A013610 → D → basis A013611
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
- A027465: Cube of lower triangular normalized binomial matrix.
- A038330 → D
- A136572 → D → basis A013610
- A038330 → reverse
- A130123 → D → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013610 → D
- A120909 → x ↦ x/(x+1)
- basis-map coefficients of A030528, shifted → θ+1 → basis A013610
- A027465 → D → D
- basis-map coefficients of A030528, shifted → D → basis A013610
- basis-map coefficients of A026729, shifted → basis A013610 → D
- A038325 → basis A007318 → basis A007318
- A085880 → x ↦ x+1 → x ↦ x+1
- basis-map coefficients of A026729, shifted → basis A013610
- A013610 → basis A013611 → reverse
- A178820 → basis A013611 → x ↦ x/(x+1)
- A131431 → D → basis A013610
- A085880 → basis A013611 → x ↦ x/(x+1)
- A136521 → basis A013610
- image under θ+1 → (θ+1)² → basis A013610
- A013610 → D → D
- A178822 → x ↦ x+1 → x ↦ x+1
- A038279 → basis A013610 → reverse
- basis-map coefficients of A030528, shifted → basis A013610 → reverse
- A013610 → basis A013611 → D
- A130123 → basis A013610 → D
- A129186 → basis A013610
- A131431 → θ+1 → basis A013610
- A136572 → basis A013610 → basis A013611
- basis-map coefficients of A026729 → basis A013610
- A120909 → basis A013610 → x ↦ x/(x+1)
- basis-map coefficients of A021012 → basis A013610
- A131431 → basis A013610
- A178820 → basis A007318 → basis A007318
- A356546 → x ↦ x+1 → x ↦ x+1
- basis-map coefficients of A112227 → θ+1 → basis A013610
- A196347 → x ↦ x+1 → basis A007318
- A013610: Triangle of coefficients in expansion of (1+3*x)^n.
- A038221 → basis A013611 → x ↦ x/(x+1)
- A038338 → x ↦ x+1 → basis A007318
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
- A130123 → (θ+1)² → basis A013610
- A038243 → basis A013609 → D
- basis-map coefficients of A026729, shifted → D → basis A013610
- basis-map coefficients of A112227 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013610 → basis A013611
- basis-map coefficients of A026729 → basis A013610 → D
- A147716 → basis A013610 → basis A013609
- A156992 → basis A007318 → x ↦ x+1
- A356546 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013610 → reverse
- A120910 → basis A007318 → reverse
- A013610 → basis A013611
- basis-map coefficients of A112227 → basis A013610 → reverse
- A131436 → basis A013610
- basis-map coefficients of A112227 → basis A013610
- A085881 → basis A013611 → x ↦ x/(x+1)
- basis-map coefficients of A026729, 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.