basis-map coefficients of A026729 → (θ+1)² → (θ+1)²

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Triangle begins:
   1: 1;
   2: 0, 16;
   3: 0,  0, 81;
   4: 0,  0,  0, 256;
   5: 0,  0,  0,   0, 625;
   6: 0,  0,  0,   0,   0, 1296;
   7: 0,  0,  0,   0,   0,    0, 2401;
   8: 0,  0,  0,   0,   0,    0,    0, 4096;
   9: 0,  0,  0,   0,   0,    0,    0,    0, 6561;
  10: 0,  0,  0,   0,   0,    0,    0,    0,    0, 10000;

The row polynomials satisfy (n^4-4*n^3+6*n^2-4*n+1)*P(n,x) = n^4*x*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(n^4*x*P[n - 1, x])/(n^4 - 4*n^3 + 6*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n^4*x*v[i - 1])/(n^4 - 4*n^3 + 6*n^2 - 4*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • 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 basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A026729 → (θ+1)² → D
• basis-map coefficients of A297191, shifted → θ+1
• A003506 → γ
• A131436 → D
• A131431 → D → D
• A131431 → (θ+1)²
• A178820 → γ
• A131436 → (θ+1)²
• A131436 → θ+1
• A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.
• A130123 → (θ+1)²
• A130123 → (θ+1)² → (θ+1)²
• A131431 → θ+1 → D
• A127648 → (θ+1)² → (θ+1)²
• basis-map coefficients of A030528, shifted → D
• basis-map coefficients of A297191, shifted → D → θ+1
• A127648 → D
• basis-map coefficients of A026729, shifted → D → D
• A132440 → θ+1 → (θ+1)²
• A130123 → D
• A038325 → γ
• basis-map coefficients of A030528, shifted → θ+1 → D
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• A134309: Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).
• A133080 → Veronese 2
• A132440 → θ+1 → D
• image under θ+1 → D
• A129186: Right shift operator generating 1's in shifted spaces.
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• A136572 → D
• basis-map coefficients of A297191, shifted → D
• basis-map coefficients of A112227 → θ+1
• A130123 → (θ+1)² → D
• A136572 → D → D
• A127648 → (θ+1)²
• basis-map coefficients of A026729, shifted → D → θ+1
• A131431 → θ+1 → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → D
• A131436 → reverse
• A131431 → D
• A132774 → Veronese 2 (j=1) → (θ+1)²
• veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• basis-map coefficients of A297191, shifted → (θ+1)² → reverse
• basis-map coefficients of A026729, shifted → (θ+1)²
• A136572 → (θ+1)²
• A130123 → D → (θ+1)²
• basis-map coefficients of A030528, shifted → (θ+1)² → θ+1
• basis-map coefficients of A026729 → (θ+1)²
• A178822 → γ
• A132774 → Veronese 2 (j=1)
• A038221 → γ
• A178821 → γ
• basis-map coefficients of A021012 → θ+1 → reverse
• A073424: Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A127773 → D → θ+1
• basis-map coefficients of A026729, shifted → θ+1
• image under θ+1 → reverse
• A132774 → Veronese 2 → θ+1
• basis-map coefficients of A021012 → θ+1
• basis-map coefficients of A026729, shifted → (θ+1)² → θ+1
• basis-map coefficients of A297191, shifted → (θ+1)²
• Hadamard-factor parent of A127773 → D
• A028326 → γ
• basis-map coefficients of A297191, shifted → (θ+1)² → D
• A127773 → θ+1 → D
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A132774 → Veronese 2 → D
• image under θ+1 → (θ+1)²
• A127773 → reverse
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• A131431 → (θ+1)² → reverse
• basis-map coefficients of A026729, shifted → θ+1 → reverse
• basis-map coefficients of A026729, shifted → D
• A130123 → D → D
• basis-map coefficients of A026729 → D → (θ+1)²
• A131431 → θ+1
• A073423: Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.
• image under θ+1 → θ+1 → D → θ+1
• basis-map coefficients of A297191, shifted → (θ+1)² → (θ+1)²
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• A127648 → (θ+1)² → D
• basis-map coefficients of A026729 → (θ+1)² → θ+1
• A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.
• A127773 → D
• basis-map coefficients of A scaled Hermite triangle.
• basis-map coefficients of A297191, shifted → D → D
• A127648 → D → (θ+1)²
• A038338 → γ
• A127773 → θ+1
• basis-map coefficients of A297191, shifted → θ+1 → D
• basis-map coefficients of A021012 → θ+1 → D
• A196347 → γ
• A136572 → θ+1 → D
• basis-map coefficients of A021012 → D
• A131431: 3n + 1 preceded by n zeros.
• A136572 → (θ+1)² → θ+1
• basis-map coefficients of A112227 → (θ+1)² → (θ+1)²
• basis-map coefficients of A030528, shifted → D → (θ+1)²
• basis-map coefficients of A030528, shifted → (θ+1)² → reverse
• basis-map coefficients of A030528, shifted → (θ+1)² → D
• basis-map coefficients of A021012 → reverse
• image under θ+1 → (θ+1)² → D
• basis-map coefficients of A026729, shifted → (θ+1)² → reverse
• A127773 → (θ+1)² → reverse
• hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A134309 → D
• basis-map coefficients of A030528, shifted → D → D
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• Hadamard-factor parent of A127773 → reverse
• A127773 → θ+1 → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → (θ+1)²
• basis-map coefficients of A021012 → (θ+1)²
• basis-map coefficients of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• A131431 → (θ+1)² → D
• basis-map coefficients of A112227 → (θ+1)² → reverse
• basis-map coefficients of A030528, shifted → (θ+1)²
• A131431 → (θ+1)² → θ+1
• A136572 → (θ+1)² → (θ+1)²
• A132774 → Veronese 2 (j=1) → θ+1
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• basis-map coefficients of A026729 → D → θ+1
• basis-map coefficients of A112227 → (θ+1)² → θ+1
• A156992 → γ
• basis-map coefficients of A297191, shifted → D → (θ+1)²
• basis-map coefficients of A112227 → θ+1 → D
• basis-map coefficients of A112227 → D
• A130123 → reverse
• A131431 → D → θ+1
• image under θ+1 → (θ+1)² → θ+1
• image under θ+1 → θ+1 → reverse
• image under θ+1 → θ+1 → (θ+1)²
• basis-map coefficients of A030528, shifted → (θ+1)² → (θ+1)²
• basis-map coefficients of A026729, shifted → θ+1 → D
• A132774 → Veronese 2
• A136572 → (θ+1)² → reverse
• basis-map coefficients of A112227 → (θ+1)² → D
• A127648 → D → θ+1
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• basis-map coefficients of A112227 → θ+1 → reverse
• A130123 → (θ+1)² → θ+1
• theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A135838 → γ
• A132774 → Veronese 2 (j=1) → D
• A356546 → γ
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A127648 → (θ+1)² → θ+1
• basis-map coefficients of A030528, shifted → θ+1 → reverse
• basis-map coefficients of A030528, shifted → reverse
• basis-map coefficients of A026729, shifted → reverse
• A344565 → γ
• A130123 → θ+1
• basis-map coefficients of A026729, shifted → D → (θ+1)²
• A127648 → D → D
• A131431 → reverse
• A136572 → θ+1
• A130123 → θ+1 → reverse
• A136572 → D → (θ+1)²
• A132774 → Veronese 2 → (θ+1)²
• A129184: Shift operator, right.
• A130123 → (θ+1)² → reverse
• A136572 → D → θ+1
• A132440 → θ+1
• basis-map coefficients of A297191, shifted → (θ+1)² → θ+1
• A085881 → γ
• basis-map coefficients of A030528, shifted → D → θ+1
• image under θ+1 → θ+1 → D → D
• A130123 → D → θ+1
• basis-map coefficients of A026729 → D → D
• basis-map coefficients of A026729 → (θ+1)² → reverse
• A127773 → (θ+1)²
• image under θ+1 → (θ+1)² → reverse
• basis-map coefficients of A112227 → (θ+1)²
• basis-map coefficients of A021012 → θ+1 → (θ+1)²
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• A136521 → D
• image under θ+1 → θ+1 → D
• A131431 → (θ+1)² → (θ+1)²
• A130123 → θ+1 → D
• basis-map coefficients of A026729 → D
• Hadamard-factor parent of A127773 → θ+1
• A127773 → D → D
• basis-map coefficients of A030528, shifted → θ+1
• A136572 → (θ+1)² → D
• Hadamard-factor parent of A127773 → (θ+1)²

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.