basis-map coefficients of A021012 → x ↦ x+1

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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:
   1:       1;
   2:      -1,       -1;
   3:       2,        4,         2;
   4:      -6,      -18,       -18,        -6;
   5:      24,       96,       144,        96,        24;
   6:    -120,     -600,     -1200,     -1200,      -600,      -120;
   7:     720,     4320,     10800,     14400,     10800,      4320,       720;
   8:   -5040,   -35280,   -105840,   -176400,   -176400,   -105840,    -35280,     -5040;
   9:   40320,   322560,   1128960,   2257920,   2822400,   2257920,   1128960,    322560,    40320;
  10: -362880, -3265920, -13063680, -30481920, -45722880, -45722880, -30481920, -13063680, -3265920, -362880;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 + (n - 1)*x)*P[n - 1, x]];
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 - 1 + (n - 1)*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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-de35ff8f60015c8a
• Coefficients of one sign. Every nonzero row, n >= 1: 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-5f6cf10d49e0d311, e-de35ff8f60015c8a
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-5f6cf10d49e0d311, e-de35ff8f60015c8a
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5f6cf10d49e0d311

Proof chains

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

• basis_map A007318 e-de35ff8f60015c8a

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-51f2fe49bf45c5d7 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-bb5cb95fed86aa9c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)
• preserver_image affine, shift 0 e-5f6cf10d49e0d311

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)
    • ↵ preserver_image mobius, shift 0 e-bb5cb95fed86aa9c (expanded above)

Related sequences ?

Same fingerprint (59)

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

• A120910 → basis A013610
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• A085880 → D
• A085881 → basis A013609
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
• A178822 → basis A013609
• basis-map coefficients of A113214, shifted → D
• A356546 → basis A013609
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
• basis-map coefficients of A026729, shifted → basis A007318
• A344565 → D
• Riordan array (1-x,x/(1-x)).
• A178821 → D
• A038208 → D
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
• A027465 → basis A013611
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
• A010054 → reverse → x ↦ x+1
• A007318 → D → D
• A178821 → basis A013609
• A007318 → ×(1+2n+n²) → D
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
• basis-map coefficients of A026729 → basis A007318
• A135838 → basis A013609
• A216973: Exponential Riordan array [x*exp(x),x].
• A085880 → basis A013609
• A156992 → basis A013609
• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
• A028326 → D
• A135838 → D
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
• basis-map coefficients of A113214, shifted → basis A013609
• A344565 → basis A013609
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
• A178822 → D
• A178820 → D
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• A038221 → basis A013609
• A135065: A127733 * A007318 as infinite lower triangular matrices.
• A007318: Binomial
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
• basis-map coefficients of A112227 → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
• A196347 → basis A013609
• A007318 → ×(1+(3/2)n+(1/2)n²) → D
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
• A028326 → basis A013609
• A038338 → basis A013609
• A038325 → basis A013609
• A178820 → basis A013609
• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
• A038330 → basis A013611