A038334 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:         64,          192,           144;
  1:        512,         2304,          3456,          1728;
  2:       4096,        24576,         55296,         55296,          20736;
  3:      32768,       245760,        737280,       1105920,         829440,         248832;
  4:     262144,      2359296,       8847360,      17694720,       19906560,       11943936,        2985984;
  5:    2097152,     22020096,      99090432,     247726080,      371589120,      334430208,      167215104,       35831808;
  6:   16777216,    201326592,    1056964608,    3170893824,     5945425920,     7134511104,     5350883328,     2293235712,      429981696;
  7:  134217728,   1811939328,   10871635968,   38050725888,    85614133248,   128421199872,   128421199872,    82556485632,    30958682112,     5159780352;
  8: 1073741824,  16106127360,  108716359680,  434865438720,  1141521776640,  2054739197952,  2568423997440,  2201506283520,  1238347284480,   412782428160,   61917364224;
  9: 8589934592, 141733920768, 1063004405760, 4783519825920, 14350559477760, 30136174903296, 45204262354944, 48433138237440, 36324853678080, 18162426839040, 5448728051712, 743008370688;

The row polynomials satisfy P(n,x) = (8+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 192*x + 144*x^2.

P[0, x_] := 64 + 192*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 192*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 12*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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 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)*8^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

Related ?

Same fingerprint ?

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