A038325 → basis A013611

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Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:             1;
  1:            22,             44;
  2:           484,           1936,            1936;
  3:         10648,          63888,          127776,           85184;
  4:        234256,        1874048,         5622144,         7496192,          3748096;
  5:       5153632,       51536320,       206145280,       412290560,        412290560,        164916224;
  6:     113379904,     1360558848,      6802794240,     18140784640,      27211176960,      21768941568,       7256313856;
  7:    2494357888,    34921010432,    209526062592,    698420208640,    1396840417280,    1676208500736,    1117472333824,     319277809664;
  8:   54875873536,   878013976576,   6146097836032,  24584391344128,   61460978360320,   98337565376512,   98337565376512,   56192894500864,   14048223625216;
  9: 1207269217792, 21730845920256, 173846767362048, 811284914356224, 2433854743068672, 4867709486137344, 6490279314849792, 5563096555585536, 2781548277792768, 618121839509504;

The row polynomials satisfy P(n,x) = (22+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(22 + 44*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (22 + 44*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-6a2839a9935a683b
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-6a2839a9935a683b
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-6a2839a9935a683b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A013611 e-6a2839a9935a683b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f7065dbe93a909bc root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-cbffb05414f7627e root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (67)

Equal up to content, sign, reversal and a small index shift.

• A085880 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A159854 → x ↦ x/(x+1)
• A038208 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A013609 → basis A013610
• A085881 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013609
• basis-map coefficients of A026729 → basis A013609
• A356546 → basis A013611
• A038338 → basis A013611
• A178820 → basis A007318
• A178821 → basis A007318
• A356546 → basis A007318
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A027466 → basis A013611
• A178821 → basis A013611
• basis-map coefficients of A112227 → basis A013609
• A120909 → D
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A178821 → x ↦ x/(x+1)
• A178822 → x ↦ x/(x+1)
• A085880 → x ↦ x/(x+1)
• A038207 → D
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A085880 → x ↦ x+1
• A344565 → x ↦ x/(x+1)
• A120910 → D
• A013609 → D
• A038221 → basis A013611
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• basis-map coefficients of A021012 → basis A013609
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A085881 → basis A013611
• A196347 → basis A013611
• A120909 → basis A013610
• A038338 → basis A007318
• A156992 → basis A007318
• A038208 → x ↦ x/(x+1)
• A344565 → basis A013611
• A135838 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A007318 → D → x ↦ x/(x+1)
• A196347 → basis A007318
• A135838 → x ↦ x/(x+1)
• A178822 → x ↦ x+1
• A028326 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A028326 → basis A007318
• A130123 → basis A013609
• A178820 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → basis A013611
• A178820 → basis A013611
• A156992 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038243 → basis A013610
• A038325 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013609
• A344565 → basis A007318
• A178822 → basis A013611
• A028326 → x ↦ x/(x+1)
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A135838 → x ↦ x+1
• A038330 → basis A013609
• A038208 → basis A007318
• A136572 → basis A013609
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…

References

1. [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