A038330 → basis A013609 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:            256,             768,              576;
  1:           4096,           18432,            27648,            13824;
  2:          65536,          393216,           884736,           884736,            331776;
  3:        1048576,         7864320,         23592960,         35389440,          26542080,           7962624;
  4:       16777216,       150994944,        566231040,       1132462080,        1274019840,         764411904,         191102976;
  5:      268435456,      2818572288,      12683575296,      31708938240,       47563407360,       42807066624,       21403533312,        4586471424;
  6:     4294967296,     51539607552,     270582939648,     811748818944,     1522029035520,     1826434842624,     1369826131968,      587068342272,      110075314176;
  7:    68719476736,    927712935936,    5566277615616,   19481971654656,    43834436222976,    65751654334464,    65751654334464,    42268920643584,    15850845241344,     2641807540224;
  8:  1099511627776,  16492674416640,  111325552312320,  445302209249280,  1168918299279360,  2104052938702848,  2630066173378560,  2254342434324480,  1268067619307520,   422689206435840,    63403380965376;
  9: 17592186044416, 290271069732864, 2177033022996480, 9796648603484160, 29389945810452480, 61718886201950208, 92578329302925312, 99191067110277120, 74393300332707840, 37196650166353920, 11158995049906176, 1521681143169024;

The row polynomials satisfy (1+x)*P(n,x) = (16+40*x+24*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 768*x + 576*x^2.

P[0, x_] := 256 + 768*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 40*x + 24*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] = 256 + 768*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 40*x + 24*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (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 ?

Same fingerprint ?

• A216973 → basis A013610
• A085881 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A085880 → D → basis A013610
• A028326 → basis A013610
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A038338 → basis A013609 → basis A013610
• A038291 → basis A013609 → basis A013611
• A027466 → basis A013611 → x ↦ x/(x+1)
• A085880 → basis A013609 → basis A013610
• A038330 → basis A013611 → basis A013610
• A135089 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A178820 → basis A013610
• A178820 → basis A013610 → D
• A178822 → basis A013610 → D
• A120910 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A136572 → basis A013610 → basis A013609
• A178820 → basis A007318 → basis A013609
• A028326 → D → basis A013610
• basis-map coefficients of A113214, shifted → D → basis A013610
• A135065 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A356546 → basis A013609 → basis A013610
• A085880 → basis A013610
• A344565 → basis A013610
• A038243 → basis A013611 → D
• A356546 → basis A013611 → basis A007318
• A085881 → basis A007318 → basis A013609
• A038221 → basis A013610 → D
• A376827 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A013610 → D → basis A013609
• A013610 → basis A013611 → basis A013609
• A038221 → basis A013610
• A038334 → reverse
• A085880 → basis A013610 → reverse
• A038208 → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A013609 → basis A013610 → x ↦ x+1
• A131431 → x ↦ x+1 → basis A013610
• A178821 → basis A013611 → basis A007318
• A134346 → basis A013610
• A178821 → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A028326 → basis A013609 → basis A013610
• A156992 → basis A013609 → basis A013610
• A156992 → basis A013610
• A094305 → basis A013610
• A028326 → basis A007318 → basis A013609
• A038338 → basis A013610 → reverse
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A038330 → basis A013609 → basis A013609
• A027465 → basis A013611 → basis A013610
• A134058 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A120910 → x ↦ x/(x+1)
• A134059 → basis A013610
• A120910 → basis A013609 → D
• A178822 → basis A013610
• A038208 → D → basis A013610
• A258758 → basis A013610
• A178822 → basis A013611 → basis A007318
• A178821 → basis A013610 → D
• A038325 → basis A013609 → basis A013610
• A038334 → D
• A356546 → x ↦ x+1 → basis A013609
• A028326 → basis A013610 → D
• A120910 → basis A013609 → reverse
• A120910 → basis A013610 → basis A013610
• A085880 → basis A013610 → D
• A038325 → basis A007318 → basis A013609
• A124860 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A085881 → basis A013609 → basis A013610
• A038221 → basis A013611 → x ↦ x+1
• A038338 → basis A013610
• A038208 → basis A013610 → D
• A136572 → basis A013609 → x ↦ x+1
• A003506 → D → basis A013610
• A085881 → basis A013611 → basis A007318
• A178820 → basis A013609 → basis A013610
• A038338 → basis A013610 → D
• A178821 → basis A013610 → reverse
• A028326 → basis A013611 → x ↦ x+1
• A132200 → basis A013610
• A196347 → basis A013610 → D
• A178821 → basis A013610
• A003506 → x ↦ x+1 → basis A013609
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A196347 → basis A013610
• A356546 → basis A013610
• A196347 → basis A013611 → x ↦ x+1
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A038221 → basis A013609 → basis A013610
• A135065 → x ↦ x+1 → basis A013609
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038325 → basis A013610
• A156992 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A178820 → basis A013611 → basis A007318
• A038243 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A038328 → basis A013609 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A178822 → x ↦ x+1 → basis A013609
• A120910 → D → basis A013609
• A013620 → D
• A156992 → basis A013611 → x ↦ x+1
• A135838 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• basis-map coefficients of A127647 → basis A013610
• A038220 → D
• A156992 → basis A007318 → basis A013609
• A178822 → basis A013609 → basis A013610

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.