Real-rooted atlas

A038323 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + 9\,n x\right)\,P_{n}(x) = \left(200 + 360\,x + 162\,x^{2} + 180\,n x + 162\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1203656
240014409722812
38000432005832023328132848
41600001152000233280018662405248806035920
532000002880000077760000933120005248800011337408266897408
664000000691200000233280000037324800003149280000136048896023808556811568334528
71280000000161280000006531840000013063680000014696640000095234227200333319795204897760256493793566976
825600000000368640000000174182400000041803776000005878656000000507915878400026665583616007836416409609917964518420823636031744
951200000000082944000000004478976000000012541132800000021163161600000022856214528000015999350169600070527747686400178523361331201983592903680869558427699200
101024000000000018432000000000011197440000000003583180800000000705438720000000091424858112000007999675084800000470184984576000017852336133120003967185807360003927513949286436017110075300864
11204800000000000405504000000000027371520000000000985374720000000002217093120000000003352244797440000003519857037312000002586017415168000001309171316428800004363904388096000086405306884300807712209209507841481657996125220864
12409600000000000088473600000000000656916480000000000262766592000000000066512793600000000001149341073408000000014079428149248000000124128835928064000007855027898572800000349112351047680000010368636826116096001850930210281881601503880795854028860597300756782338048
138192000000000000019169280000000000001552711680000000000068319313920000000000192148070400000000000373535848857600000000522950188400640000000537891622354944000000408461450725785600000226923028180992000000898615191596728320002406209273366446080039100900692204748802915215081193963522465880711110638764032
141638400000000000000412876800000000000003622993920000000000001739037081600000000000538014597120000000000011621115297792000000000183032565940224000000002151566489419776000000019061534367203328000000127076895781355520000006290306341177098240000224579532180868300800054741260969086648320081626022273430978560562220051373121536099904431751614849925120
Technical details
Node id
n-8785b7a86fcc7953 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-acd18755d266097d

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:           20,            36;
  2:          400,          1440,            972;
  3:         8000,         43200,          58320,           23328;
  4:       160000,       1152000,        2332800,         1866240,          524880;
  5:      3200000,      28800000,       77760000,        93312000,        52488000,        11337408;
  6:     64000000,     691200000,     2332800000,      3732480000,      3149280000,      1360488960,       238085568;
  7:   1280000000,   16128000000,    65318400000,    130636800000,    146966400000,     95234227200,     33331979520,     4897760256;
  8:  25600000000,  368640000000,  1741824000000,   4180377600000,   5878656000000,   5079158784000,   2666558361600,   783641640960,    99179645184;
  9: 512000000000, 8294400000000, 44789760000000, 125411328000000, 211631616000000, 228562145280000, 159993501696000, 70527747686400, 17852336133120, 1983592903680;
%F Formula
The row polynomials satisfy (10+9*n*x)*P(n,x) = (200+(180*n+360)*x+(162*n+162)*x^2)*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[Together[((200 + (180*n + 360)*x + (162*n + 162)*x^2)*P[n - 1, x])/(10 + 9*n*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] = ((200 + (180*n + 360)*x + (162*n + 162)*x^2)*v[i - 1])/(10 + 9*n*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

Proof chains ? JSON

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.