A038329 → x ↦ x+1 → x ↦ x/(x+1)

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           225,             540,             324;
  1:          3375,           12150,           14580,             5832;
  2:         50625,          243000,          437400,           349920,           104976;
  3:        759375,         4556250,        10935000,         13122000,          7873200,          1889568;
  4:      11390625,        82012500,       246037500,        393660000,        354294000,        170061120,          34012224;
  5:     170859375,      1435218750,      5166787500,      10333575000,      12400290000,       8928208800,        3571283520,         612220032;
  6:    2562890625,     24603750000,    103335750000,     248005800000,     372008700000,     357128352000,      214277011200,       73466403840,      11019960576;
  7:   38443359375,    415188281250,   1992903750000,    5580130500000,   10044234900000,   12053081880000,     9642465504000,     4958982259200,    1487694677760,     198359290368;
  8:  576650390625,   6919804687500,  37366945312500,  119574225000000,  251105872500000,  361592456400000,   361592456400000,   247949112960000,  111577100832000,   29753893555200,   3570467226624;
  9: 8649755859375, 114176777343750, 685060664062500, 2466218390625000, 5918924137500000, 9943792551000000, 11932551061200000, 10227900909600000, 6136740545760000, 2454696218304000, 589127092392960, 64268410079232;

The row polynomials satisfy (1+x)*P(n,x) = (15+33*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 540*x + 324*x^2.

P[0, x_] := 225 + 540*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15 + 33*x + 18*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] = 225 + 540*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15 + 33*x + 18*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 basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • 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
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A013612 → x ↦ x+1 → reverse
• A038338 → basis A013610 → basis A013609
• A013612 → x ↦ x+1 → D
• A147716 → basis A013610 → x ↦ x/(x+1)
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A038291 → basis A013609 → x ↦ x/(x+1)
• A038259 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A178821 → basis A013610 → basis A013609
• A038255 → basis A013610 → basis A013611
• A038220 → basis A013610 → D
• A196347 → basis A013610 → basis A013609
• A120910 → basis A013609 → basis A013610
• A156992 → basis A013610 → basis A013609
• A038334 → basis A013610
• A178822 → basis A013610 → basis A013609
• A038336 → D
• A038271 → basis A013609
• A038334 → basis A013610 → D
• A085880 → basis A013610 → basis A013609
• A038334 → basis A013610 → reverse
• A038314 → D
• A038221 → basis A013610 → basis A013609
• A013612 → x ↦ x+1
• A178820 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038220 → basis A013610
• A038271 → basis A013609 → D
• A028326 → basis A013610 → basis A013609

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.