A038334 → D

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No OEIS matches.

Submission text ?

Triangle begins:
  0:           192,            128;
  1:          3456,           4608,            1536;
  2:         55296,         110592,           73728,           16384;
  3:        829440,        2211840,         2211840,          983040,          163840;
  4:      11943936,       39813120,        53084160,        35389440,        11796480,         1572864;
  5:     167215104,      668860416,      1114767360,       990904320,       495452160,       132120576,        14680064;
  6:    2293235712,    10701766656,     21403533312,     23781703680,     15854469120,      6341787648,      1409286144,      134217728;
  7:   30958682112,   165112971264,    385263599616,    513684799488,    428070666240,    228304355328,     76101451776,    14495514624,    1207959552;
  8:  412782428160,  2476694568960,   6604518850560,  10273695989760,  10273695989760,   6849130659840,   3044058071040,   869730877440,  144955146240,   10737418240;
  9: 5448728051712, 36324853678080, 108974561034240, 193732552949760, 226021311774720, 180817049419776, 100453916344320, 38268158607360, 9567039651840, 1417339207680, 94489280512;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(8*n+16)*x)*P(n-1,x) for n >= 1, with P(0,x) = 192 + 128*x.

P[0, x_] := 192 + 128*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (8*n + 16)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 192 + 128*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (8*n + 16)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 ?

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