A038330 → x ↦ x/(x+1)

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:          144,            384,            256;
  1:         1728,           6912,           9216,            4096;
  2:        20736,         110592,         221184,          196608,           65536;
  3:       248832,        1658880,        4423680,         5898240,         3932160,          1048576;
  4:      2985984,       23887872,       79626240,       141557760,       141557760,         75497472,         16777216;
  5:     35831808,      334430208,     1337720832,      2972712960,      3963617280,       3170893824,       1409286144,        268435456;
  6:    429981696,     4586471424,    21403533312,     57076088832,     95126814720,     101468602368,      67645734912,      25769803776,       4294967296;
  7:   5159780352,    61917364224,   330225942528,   1027369598976,   2054739197952,    2739652263936,    2435246456832,    1391569403904,     463856467968,     68719476736;
  8:  61917364224,   825564856320,  4953389137920,  17612050268160,  41094783959040,   65751654334464,   73057393704960,   55662776156160,   27831388078080,   8246337208320,   1099511627776;
  9: 743008370688, 10897456103424, 72649707356160, 290598829424640, 774930211799040, 1446536395358208, 1928715193810944, 1836871613153280, 1224581075435520, 544258255749120, 145135534866432, 17592186044416;

The row polynomials satisfy (1+x)*P(n,x) = (12+28*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 384*x + 256*x^2.

P[0, x_] := 144 + 384*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 28*x + 16*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 384*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 28*x + 16*x^2)*v[i - 1])/(1 + 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Candidate equalities ?

• A038330 → basis A013610 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A120909 → basis A013610 → basis A013609
• A178822 → basis A013611 → basis A013609
• A130123 → basis A013610 → x ↦ x+1
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• A013610 → basis A013611 → x ↦ x+1
• A356546 → x ↦ x+1 → basis A013611
• A120910 → D → basis A013611
• A038329 → basis A013610 → basis A013609
• A178822 → x ↦ x+1 → basis A013611
• A178821 → basis A013611 → basis A013609
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A038330 → basis A013610
• A130123 → basis A013609 → basis A013609
• A120910 → basis A013611
• A038221 → basis A013611 → basis A013609
• A027465 → D → basis A013610
• A038335 → reverse
• A038243 → basis A013609 → x ↦ x/(x+1)
• A027465 → x ↦ x/(x+1)
• A038279 → basis A013610 → x ↦ x/(x+1)
• A085881 → basis A013611 → basis A013609
• A085880 → x ↦ x+1 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A003506 → x ↦ x+1 → basis A013611
• A196347 → basis A013611 → basis A013609
• A178820 → basis A007318 → basis A013611
• A038330 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A028326 → basis A013611 → basis A013609
• A094305 → x ↦ x+1 → basis A013611
• A356546 → basis A013611 → basis A013609
• A013609 → basis A013609
• A013609 → D → basis A013609
• A038284 → reverse
• A038335 → D
• A178821 → basis A007318 → basis A013611
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A178820 → basis A013611 → basis A013609
• A038334 → basis A013609 → basis A013610
• A120910 → basis A013611 → D
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A038329 → basis A013611 → basis A013610
• A013610 → basis A007318 → D
• A028326 → basis A007318 → basis A013611
• A120910 → basis A013611 → reverse
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A156992 → basis A013611 → basis A013609
• A136572 → basis A013610 → basis A007318
• A196347 → x ↦ x+1 → basis A013611
• A135065 → x ↦ x+1 → basis A013611
• A013609 → basis A013609 → D
• A038208 → basis A013611 → basis A013609
• A013610 → basis A007318
• A085880 → basis A013611 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A038284 → D

References

1. [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
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.