Real-rooted atlas

A038323 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(20 + 27\,n x\right)\,P_{n}(x) = \left(400 + 1080\,x + 729\,x^{2} + 540\,n x + 729\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1205474
2400216021874747
380006480013122078732282752
4160000172800052488006298560265720516092565
532000004320000017496000031492800026572050086093442888101942
66400000010368000005248800000125971200001594323000010331213040271194342347933106463
7128000000024192000000146966400000440899200000744017400000723184912800379672079220836828256242543894817644
825600000000552960000000391910400000014108774400000297606960000003856986201600030373766337600133892520998402541865828329133241880681769
9512000000000124416000000001007769600000004232632320000001071385056000000173564379072000018224259802560001205032688985600457535849099220762559748498706905273131910690
10102400000000002764800000000002519424000000000120932352000000003571283520000000069425751628800000911212990128000008033551259904000045753584909922000152511949699740002264802453041139354764359973577139
112048000000000006082560000000000615859200000000003325639680000000001122403392000000000254561089305600000040093371565632000004418453192947200000335526289339428000016776314466971400004982565396690505806670872679866627618094101489125536856
12409600000000000013271040000000000014780620800000000008868372480000000000336721017600000000008727808776192000000016037348626252800000021208575326146560000020131577360365680000013421051573577120000059790784760286069600160100944316799062401951230258860988573917171068796168564413
1381920000000000000287539200000000000034936012800000000000230577684480000000000972749606400000000000283653785226240000000059567294897510400000009190382641330176000000104684202273901536000008723683522825128000000518186801255812603200020813122761183878112005073198673038570289805673577214226566773846244210277361534139918
1416384000000000000006193152000000000000081517363200000000000058692501504000000000002723698897920000000000088247844292608000000000208485532141286400000000367615305653207040000000488526277278207168000000488526277278207168000000362730760879068822240000194255812437716195712000710247814225399840572001588601619983438696664016412848369726853881352320924875101247850363975
Technical details
Node id
n-e32c634c0eedfe34 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b410ed18e0f5c25b

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,             54;
  2:          400,           2160,            2187;
  3:         8000,          64800,          131220,           78732;
  4:       160000,        1728000,         5248800,         6298560,          2657205;
  5:      3200000,       43200000,       174960000,       314928000,        265720500,         86093442;
  6:     64000000,     1036800000,      5248800000,     12597120000,      15943230000,      10331213040,       2711943423;
  7:   1280000000,    24192000000,    146966400000,    440899200000,     744017400000,     723184912800,     379672079220,      83682825624;
  8:  25600000000,   552960000000,   3919104000000,  14108774400000,   29760696000000,   38569862016000,   30373766337600,   13389252099840,   2541865828329;
  9: 512000000000, 12441600000000, 100776960000000, 423263232000000, 1071385056000000, 1735643790720000, 1822425980256000, 1205032688985600, 457535849099220, 76255974849870;
%F Formula
The row polynomials satisfy (20+27*n*x)*P(n,x) = (400+(540*n+1080)*x+(729*n+729)*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[((400 + (540*n + 1080)*x + (729*n + 729)*x^2)*P[n - 1, x])/(20 + 27*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] = ((400 + (540*n + 1080)*x + (729*n + 729)*x^2)*v[i - 1])/(20 + 27*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.