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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"c2a631b1e361a8680a36622d058fad4aceab8a9d7bedc80c340222b0a9217d2f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-047228a38014e6d4","role":"sequence","name":"Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-bb19e463b7d0d5d4","kind":"oeis_match","node_a_id":"n-8304b53a8f961041","node_b_id":null,"fingerprint":null,"oeis_id":"A350265","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A350265","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A350265","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,     1;\n  2:   5,     6,      1;\n  3:  11,    30,     13,      1;\n  4:  21,   120,    110,     24,      1;\n  5:  36,   385,    700,    315,     40,      1;\n  6:  57,  1036,   3430,   2940,    770,     62,     1;\n  7:  85,  2436,  13524,  20580,   9996,   1680,    91,    1;\n  8: 121,  5160,  44856, 112896,  95256,  29232,  3360,  128,   1;\n  9: 166, 10065, 129690, 507276, 698544, 365904, 76230, 6270, 174, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^5+7*n^4+11*n^3-n^2+30*n+72+(3*n^4+18*n^3+9*n^2-78*n-72)*x+(3*n^3+19*n^2+22*n-24)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (n^5+10*n^4+35*n^3+56*n^2+66*n+72+(10*n^5+13*n^4-4*n^3-n^2-66*n-72)*x+(30*n^4+15*n^3-61*n^2-40*n-24)*x^2+(30*n^3+21*n^2-3*n+12)*x^3+(10*n^2+10*n)*x^4)*P(n-1,x) + ((3*n^4+36*n^3+57*n^2+24*n+72)*x-(15*n^4+3*n^3-113*n^2-32*n+84)*x^2-(45*n^3-8*n^2-155*n+24)*x^3-(45*n^2-3*n-24)*x^4-15*n*x^5)*d/dx P(n-1,x) + ((6*n^3+10*n^2+n+12)*x^2+(6*n^3+28*n^2+13*n-12)*x^3+(18*n^2+30*n-18)*x^4+(18*n+12)*x^5+6*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^5 + 10*n^4 + 35*n^3 + 56*n^2 + 66*n + 72 + (10*n^5 + 13*n^4 - 4*n^3 - n^2 - 66*n - 72)*x + (30*n^4 + 15*n^3 - 61*n^2 - 40*n - 24)*x^2 + (30*n^3 + 21*n^2 - 3*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*P[n - 1, x] + ((3*n^4 + 36*n^3 + 57*n^2 + 24*n + 72)*x - (15*n^4 + 3*n^3 - 113*n^2 - 32*n + 84)*x^2 - (45*n^3 - 8*n^2 - 155*n + 24)*x^3 - (45*n^2 - 3*n - 24)*x^4 - 15*n*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 + n + 12)*x^2 + (6*n^3 + 28*n^2 + 13*n - 12)*x^3 + (18*n^2 + 30*n - 18)*x^4 + (18*n + 12)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 7*n^4 + 11*n^3 - n^2 + 30*n + 72 + (3*n^4 + 18*n^3 + 9*n^2 - 78*n - 72)*x + (3*n^3 + 19*n^2 + 22*n - 24)*x^2 + (n^2 + 7*n + 12)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^5 + 10*n^4 + 35*n^3 + 56*n^2 + 66*n + 72 + (10*n^5 + 13*n^4 - 4*n^3 - n^2 - 66*n - 72)*x + (30*n^4 + 15*n^3 - 61*n^2 - 40*n - 24)*x^2 + (30*n^3 + 21*n^2 - 3*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*v[i - 1] + ((3*n^4 + 36*n^3 + 57*n^2 + 24*n + 72)*x - (15*n^4 + 3*n^3 - 113*n^2 - 32*n + 84)*x^2 - (45*n^3 - 8*n^2 - 155*n + 24)*x^3 - (45*n^2 - 3*n - 24)*x^4 - 15*n*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 + n + 12)*x^2 + (6*n^3 + 28*n^2 + 13*n - 12)*x^3 + (18*n^2 + 30*n - 18)*x^4 + (18*n + 12)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 7*n^4 + 11*n^3 - n^2 + 30*n + 72 + (3*n^4 + 18*n^3 + 9*n^2 - 78*n - 72)*x + (3*n^3 + 19*n^2 + 22*n - 24)*x^2 + (n^2 + 7*n + 12)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)","D+1"],"edges":["e-548c39a476f786f5","e-5336a01b25274fbd"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 D+1"},"headline":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 D+1","chain":null}