Real-rooted atlas

A038271 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(17 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1171027
2289340100729
3491386705100100019683
4835211965201734006800010000531441
5141985741760504913000289000085000010000014348907
624137569851914201252815009826000043350000102000001000000387420489
741033867316896298302981699700292323500017195500006069000001190000001000000010460353203
86975757441328270938406758519320079511992000584647000002751280000080920000001360000000100000000282429536481
911858787649762781816969014772192228002027555796000178901982000010523646000004126920000001040400000001530000000010000000007625597484987
1020159939004491185878764970031390908484500492406407600005068889490000035780396400000175394100000005895600000000130050000000017000000000010000000000205891132094649
1134271896307633221759329049390652233320733500115099997776500013541176209000001115155687800000655973934000000275619300000000810645000000001589500000000018700000000001000000000005559060566555523
12582622237229761411262755691596013305559742963400260893328293400003452999933295000032498822901600000223031137560000001124526744000000041342895000000001080860000000000190740000000000204000000000001000000000000150094635296999121
1399045780329059377574089083986893026732079119953740057657425552841400084790331695355000089777998265670000070414116286800000041420068404000000018273559590000000059717515000000000140511800000000002254200000000000221000000000000100000000000004052555153018976267
1416837782655940092913866409246068311805301862358790825100124749702559784120002018009894349449000023741292874699400000209481995953230000001408282325736000000072485119707000000002842553714000000000836045210000000000178833200000000000262990000000000002380000000000000100000000000000109418989131512359209
Technical details
Node id
n-7685734ea2b0f312 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-700950e0f523054a

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 Example
Triangle begins:
  0:            1;
  1:           17,           10;
  2:          289,          340,           100;
  3:         4913,         8670,          5100,          1000;
  4:        83521,       196520,        173400,         68000,         10000;
  5:      1419857,      4176050,       4913000,       2890000,        850000,        100000;
  6:     24137569,     85191420,     125281500,      98260000,      43350000,      10200000,      1000000;
  7:    410338673,   1689629830,    2981699700,    2923235000,    1719550000,     606900000,    119000000,     10000000;
  8:   6975757441,  32827093840,   67585193200,   79511992000,   58464700000,   27512800000,   8092000000,   1360000000,   100000000;
  9: 118587876497, 627818169690, 1477219222800, 2027555796000, 1789019820000, 1052364600000, 412692000000, 104040000000, 15300000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (17+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(17 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (17 + 10*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.