A013609 → D → basis A013609
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(3\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||||||||||
| 1 | 2 | 2 | |||||||||||||
| 2 | 12 | 16 | 28 | ||||||||||||
| 3 | 54 | 144 | 96 | 294 | |||||||||||
| 4 | 216 | 864 | 1152 | 512 | 2744 | ||||||||||
| 5 | 810 | 4320 | 8640 | 7680 | 2560 | 24010 | |||||||||
| 6 | 2916 | 19440 | 51840 | 69120 | 46080 | 12288 | 201684 | ||||||||
| 7 | 10206 | 81648 | 272160 | 483840 | 483840 | 258048 | 57344 | 1647086 | |||||||
| 8 | 34992 | 326592 | 1306368 | 2903040 | 3870720 | 3096576 | 1376256 | 262144 | 13176688 | ||||||
| 9 | 118098 | 1259712 | 5878656 | 15676416 | 26127360 | 27869184 | 18579456 | 7077888 | 1179648 | 103766418 | |||||
| 10 | 393660 | 4723920 | 25194240 | 78382080 | 156764160 | 209018880 | 185794560 | 106168320 | 35389440 | 5242880 | 807072140 | ||||
| 11 | 1299078 | 17321040 | 103926240 | 369515520 | 862202880 | 1379524608 | 1532805120 | 1167851520 | 583925760 | 173015040 | 23068672 | 6214455478 | |||
| 12 | 4251528 | 62355744 | 415704960 | 1662819840 | 4434186240 | 8277147648 | 11036196864 | 10510663680 | 7007109120 | 3114270720 | 830472192 | 100663296 | 47455841832 | ||
| 13 | 13817466 | 221079456 | 1621249344 | 7205552640 | 21616657920 | 46115536896 | 71735279616 | 81983176704 | 68319313920 | 40485519360 | 16194207744 | 3925868544 | 436207616 | 359873467226 | |
| 14 | 44641044 | 773778096 | 6190224768 | 30263321088 | 100877736960 | 242106568704 | 430411677696 | 573882236928 | 573882236928 | 425097953280 | 226718908416 | 82443239424 | 18320719872 | 1879048192 | 2712892291396 |
Compute more
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Technical details
- Node id
n-daa2820e02bfd9b6· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-c91d5d506c107735
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
- 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: 0; 1: 2; 2: 12, 16; 3: 54, 144, 96; 4: 216, 864, 1152, 512; 5: 810, 4320, 8640, 7680, 2560; 6: 2916, 19440, 51840, 69120, 46080, 12288; 7: 10206, 81648, 272160, 483840, 483840, 258048, 57344; 8: 34992, 326592, 1306368, 2903040, 3870720, 3096576, 1376256, 262144; 9: 118098, 1259712, 5878656, 15676416, 26127360, 27869184, 18579456, 7077888, 1179648;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (3*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 4*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + 4*n*x)*v[i - 1])/(n - 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 ≥ 2
Proof chains ? JSON
-
Geometric basis map (elementary) polytool ? step ›
-
from
derivative of Triangle of coefficients in expansion of (1+2*x)^n.?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
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
-
Geometric basis map (elementary) polytool ? step ›
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Gamma criterion (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1-x,x/(1-x)).?
-
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
-
from
Riordan array (1-x,x/(1-x)).?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
from
binomial?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).?
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Triangle of coefficients in expansion of (1+2*x)^n.?
-
-
from
derivative of Triangle of coefficients in expansion of (1+2*x)^n.?
Related ?
Same fingerprint ?
- A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
- A196347 → x ↦ x+1 → basis A013611
- A038221 → basis A013611 → basis A013609
- A085880 → basis A013611 → basis A013609
- A028326 → basis A007318 → basis A013611
- A013610 → basis A013611 → x ↦ x+1
- A003506 → x ↦ x+1 → basis A013611
- A196347 → basis A013611 → basis A013609
- A120909 → basis A013610 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
- A178820 → basis A013611 → basis A013609
- basis-map coefficients of A112227 → basis A013610 → basis A007318
- A038335 → D
- A130123 → basis A013609 → basis A013609
- A094305 → x ↦ x+1 → basis A013611
- A156992 → basis A013611 → basis A013609
- A085880 → x ↦ x+1 → basis A013611
- A120910 → basis A013611 → reverse
- basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
- A013610 → basis A007318
- A038329 → basis A013611 → basis A013610
- A038335 → reverse
- basis-map coefficients of A026729 → basis A013610 → basis A007318
- A038329 → basis A013610 → basis A013609
- basis-map coefficients of A026729 → basis A013609 → basis A013609
- A120910 → basis A013611 → D
- A038330 → basis A013610 → reverse
- A135065 → x ↦ x+1 → basis A013611
- A038334 → basis A013609 → basis A013610
- A356546 → x ↦ x+1 → basis A013611
- A038279 → basis A013610 → x ↦ x/(x+1)
- A038330 → x ↦ x/(x+1)
- A027465 → x ↦ x/(x+1)
- A038284 → reverse
- A178822 → basis A013611 → basis A013609
- A178822 → x ↦ x+1 → basis A013611
- A013610 → basis A007318 → D
- basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
- A028326 → basis A013611 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
- basis-map coefficients of A112227 → basis A013609 → basis A013609
- A013609 → basis A013609 → D
- A130123 → basis A013610 → x ↦ x+1
- A136572 → basis A013610 → basis A007318
- A038208 → basis A013611 → basis A013609
- A178821 → basis A013611 → basis A013609
- A085881 → basis A013611 → basis A013609
- A178821 → basis A007318 → basis A013611
- A038330 → basis A013610
- A038243 → basis A013609 → x ↦ x/(x+1)
- Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
- A013609 → basis A013609
- A027465 → D → basis A013610
- A120910 → basis A013611
- A038284 → D
- A356546 → basis A013611 → basis A013609
- A038330 → basis A013610 → D
- A120910 → D → basis A013611
- A178820 → basis A007318 → basis A013611
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [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.