Real-rooted atlas

A038324 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(41 + 40\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1414081
21681328016006561
36892120172019680064000531441
42825761110273601613760010496000256000043046721
5115856201565152200110273600010758400005248000001024000003486784401
6475010424127805488240678182640008821888000064550400000251904000004096000000282429536481
719475427388113300291874803892768353600632970464000061753216000003614822400000117555200000016384000000022876792454961
87984925229121623213676419202128046699968004152286243840005063763712000003952205824000001927905280000005373952000000065536000000001853020188851841
93273819343939612874573082483560112178461755456002553656039961600037370576194560000364590987264000002371323494400000099149414400000002418278400000000262144000000000150094635296999121
1013422659310152401130952773757584400574914616496712000149571282340608000025536560399616000002989646095564800000243060658176000000013550419968000000004957470720000000001074790400000000001048576000000000012157665459056928801
115503290317162484415905970096467056440288096102266685680008432081041951776000016452841057466880000022472173151662080000021924071367475200000015278098513920000000074527309824000000000242365235200000000004729077760000000000419430400000000000984770902183611232881
122256349030036618608126415793522379925168014174328231520935456004609537636266970880000101184972503421312000001579472741516820480000017977738521329664000000150336489376972800000009166859108352000000000397478985728000000000011633531289600000000002063597568000000000001677721600000000000079766443076872509863361
1392510310231501362932111733014956190416762120686810631581878054368002456883560130295479040005992398927147062144000001052323714035581644800000136887637598124441600000013354891472987750400000009771871809503232000000005296407484825600000000002066890725785600000000005499487518720000000000089422561280000000000006710886400000000000006461081889226673298932241
143792922719491555880216151805773729640763241976032852441877333166933936001282046512286172368153600034396369841824136706560000671148679840470960128000009821687997665428684800000010951011007849955328000000093484240310914252800000000608027579257978880000000002965988191502336000000000010522352785817600000000000256642750873600000000000038520487936000000000000026843545600000000000000523347633027360537213511521
Technical details
Node id
n-2669785079ec2366 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-3da6e3ee63c9a338

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:              41,               40;
  2:            1681,             3280,              1600;
  3:           68921,           201720,            196800,             64000;
  4:         2825761,         11027360,          16137600,          10496000,           2560000;
  5:       115856201,        565152200,        1102736000,        1075840000,         524800000,         102400000;
  6:      4750104241,      27805488240,       67818264000,       88218880000,       64550400000,       25190400000,        4096000000;
  7:    194754273881,    1330029187480,     3892768353600,     6329704640000,     6175321600000,     3614822400000,     1175552000000,     163840000000;
  8:   7984925229121,   62321367641920,   212804669996800,   415228624384000,   506376371200000,   395220582400000,   192790528000000,   53739520000000,    6553600000000;
  9: 327381934393961, 2874573082483560, 11217846175545600, 25536560399616000, 37370576194560000, 36459098726400000, 23713234944000000, 9914941440000000, 2418278400000000, 262144000000000;
%F Formula
The row polynomials satisfy P(n,x) = (41+40*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[(41 + 40*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] = (41 + 40*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

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.