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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":"bedfc14c2dbd83280c0a8ed2575887d00d4104a5ccc22115bfd19dbfb8d9ca6a","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-563143c7a3f59683","role":"sequence","name":"shift_x_plus_one of 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":[],"oeis":[],"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:       3,       1;\n  2:      12,       8,        1;\n  3:      55,      59,       16,       1;\n  4:     276,     416,      188,      28,       1;\n  5:    1477,    2895,     1895,     485,      45,      1;\n  6:    8296,   20112,    17505,    6660,    1095,     68,      1;\n  7:   48393,  140161,   153426,   79219,   19796,   2247,     98,    1;\n  8:  291010,  981808,  1300516,  857976,  296366,  52136,   4284,  136,   1;\n  9: 1794320, 6917640, 10781850, 8714370, 3903270, 964824, 125076, 7698, 183, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^5+10*n^4+32*n^3+28*n^2-19*n-12+(3*n^4+24*n^3+50*n^2-13*n-84)*x+(3*n^3+22*n^2+43*n+12)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (11*n^5+53*n^4+76*n^3+25*n^2-33*n-12+(10*n^5+73*n^4+116*n^3-20*n^2-115*n-84)*x+(30*n^4+105*n^3+62*n^2+11*n+12)*x^2+(30*n^3+61*n^2+37*n+12)*x^3+(10*n^2+10*n)*x^4)*P(n-1,x) - (12*n^4+12*n^3-133*n^2-199*n+12+(27*n^4+105*n^3-127*n^2-490*n+72)*x+(15*n^4+138*n^3+133*n^2-365*n+12)*x^2+(45*n^3+172*n^2-17*n-72)*x^3+(45*n^2+72*n-24)*x^4+15*n*x^5)*d/dx P(n-1,x) + (12*n^3+56*n^2+62*n+(30*n^3+176*n^2+251*n+12)*x+(24*n^3+202*n^2+400*n+78)*x^2+(6*n^3+100*n^2+313*n+156)*x^3+(18*n^2+120*n+132)*x^4+(18*n+48)*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[((11*n^5 + 53*n^4 + 76*n^3 + 25*n^2 - 33*n - 12 + (10*n^5 + 73*n^4 + 116*n^3 - 20*n^2 - 115*n - 84)*x + (30*n^4 + 105*n^3 + 62*n^2 + 11*n + 12)*x^2 + (30*n^3 + 61*n^2 + 37*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*P[n - 1, x] - (12*n^4 + 12*n^3 - 133*n^2 - 199*n + 12 + (27*n^4 + 105*n^3 - 127*n^2 - 490*n + 72)*x + (15*n^4 + 138*n^3 + 133*n^2 - 365*n + 12)*x^2 + (45*n^3 + 172*n^2 - 17*n - 72)*x^3 + (45*n^2 + 72*n - 24)*x^4 + 15*n*x^5)*D[P[n - 1, x], x] + (12*n^3 + 56*n^2 + 62*n + (30*n^3 + 176*n^2 + 251*n + 12)*x + (24*n^3 + 202*n^2 + 400*n + 78)*x^2 + (6*n^3 + 100*n^2 + 313*n + 156)*x^3 + (18*n^2 + 120*n + 132)*x^4 + (18*n + 48)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 32*n^3 + 28*n^2 - 19*n - 12 + (3*n^4 + 24*n^3 + 50*n^2 - 13*n - 84)*x + (3*n^3 + 22*n^2 + 43*n + 12)*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] = ((11*n^5 + 53*n^4 + 76*n^3 + 25*n^2 - 33*n - 12 + (10*n^5 + 73*n^4 + 116*n^3 - 20*n^2 - 115*n - 84)*x + (30*n^4 + 105*n^3 + 62*n^2 + 11*n + 12)*x^2 + (30*n^3 + 61*n^2 + 37*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*v[i - 1] - (12*n^4 + 12*n^3 - 133*n^2 - 199*n + 12 + (27*n^4 + 105*n^3 - 127*n^2 - 490*n + 72)*x + (15*n^4 + 138*n^3 + 133*n^2 - 365*n + 12)*x^2 + (45*n^3 + 172*n^2 - 17*n - 72)*x^3 + (45*n^2 + 72*n - 24)*x^4 + 15*n*x^5)*deriv(v[i - 1], x) + (12*n^3 + 56*n^2 + 62*n + (30*n^3 + 176*n^2 + 251*n + 12)*x + (24*n^3 + 202*n^2 + 400*n + 78)*x^2 + (6*n^3 + 100*n^2 + 313*n + 156)*x^3 + (18*n^2 + 120*n + 132)*x^4 + (18*n + 48)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 32*n^3 + 28*n^2 - 19*n - 12 + (3*n^4 + 24*n^3 + 50*n^2 - 13*n - 84)*x + (3*n^3 + 22*n^2 + 43*n + 12)*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-047228a38014e6d4","steps":["x \u21a6 x+1","D+1"],"edges":["e-0c759266bd791ca5","e-c50ddf80e741a373"],"text":"A056939 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A056939 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}