Real-rooted atlas

A038277 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(40 + 22\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1402262
2160017604843844
3640001056005808010648238328
4256000056320004646400170368023425614776336
5102400000281600000309760000170368000468512005153632916132832
640960000001351680000018585600000136294400005622144000123687168011337990456800235584
716384000000063078400000010407936000009540608000005247334400001731620352003174637312024943578883521614606208
865536000000002883584000000055508992000000610598912000004197867520000018470617088000507941969920079819452416054875873536218340105584896
92621440000000001297612800000000285474816000000036635934720000003022464614400000166235553792000060953036390400014367501434880019755314472960120726921779213537086546263552
1010485760000000000576716800000000001427374080000000002093481984000000002014976409600000001329884430336000006095303639040000019156668579840000395106289459200048290768711680026559922791424839299365868340224
114194304000000000002537553920000000000697827328000000000011514150912000000000126655660032000000009752485822464000000536386720235520000021072335437824000005794892245401600001062396911656960001168636602822656058431830141132852036560683837093888
121677721600000000000011072962560000000000033495711744000000000061408804864000000000075993396019200000000066874188496896000000042910937618841600000020229442020311040000069538706944819200000169983505865113600002804727846774374400280472784677437440128550026310492163226266762397899821056
13671088640000000000000479828377600000000000015834336460800000000000319325785292800000000004390729547776000000000043468222522982400000000318766965168537600000001753218308426956800000072320255222611968000002209785576246476800000486152826774224896000729229240161337344006684601368145592320282810057883082752200028539268669788905472
14268435456000000000000002066953011200000000000007389357015040000000000001625658543308800000000000245880854675456000000000027046894014300160000000002231368756179763200000000140257464674156544000000067498904874437836800000024749598453960540160000068061395748391485440000136122791496782970880001871688383080765849600158373632414526341120622182127342782054412401769434657526912139264
Technical details
Node id
n-b2d7d4490f3d83b7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-590b4d5bb5323b01

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:              40,               22;
  2:            1600,             1760,              484;
  3:           64000,           105600,            58080,            10648;
  4:         2560000,          5632000,          4646400,          1703680,           234256;
  5:       102400000,        281600000,        309760000,        170368000,         46851200,          5153632;
  6:      4096000000,      13516800000,      18585600000,      13629440000,       5622144000,       1236871680,       113379904;
  7:    163840000000,     630784000000,    1040793600000,     954060800000,     524733440000,     173162035200,     31746373120,      2494357888;
  8:   6553600000000,   28835840000000,   55508992000000,   61059891200000,   41978675200000,   18470617088000,   5079419699200,    798194524160,    54875873536;
  9: 262144000000000, 1297612800000000, 2854748160000000, 3663593472000000, 3022464614400000, 1662355537920000, 609530363904000, 143675014348800, 19755314472960, 1207269217792;
%F Formula
The row polynomials satisfy P(n,x) = (40+22*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[(40 + 22*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] = (40 + 22*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.