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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-77739a8efa123beb"]},{"id":191575,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-77739a8efa123beb"]},{"id":191576,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"98073ac497468b7c91743b32d31944827de66b072fc693f339a94d2c7778cd70","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-00bc3b0d8b47bf75","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:          5,           3;\n  2:         38,          84,           27;\n  3:        348,        1524,         1827,           405;\n  4:       3792,       25800,        59472,         50112,           8505;\n  5:      48288,      452400,      1593612,       2541348,        1667709,         229635;\n  6:     706176,     8489088,     40783752,      98371368,      121021614,       65367972,        7577955;\n  7:   11683584,   172131840,   1054712304,    3447065808,     6370081704,     6411103992,     2950982523,      295540245;\n  8:  215967744,  3776601600,  28245930624,  117227590848,   292165637904,   439536802032,   375414436368,   150822168480,   13299311025;\n  9: 4413837312, 89481120768, 792092805120, 4000466793600, 12619783474704, 25515760504464, 32459291153064, 24126746981688, 8609583053205, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(48*n^3-144*n^2+132*n-36)*x+(72*n^3-252*n^2+270*n-81)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(24*n^2-48*n+18)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (48*n^3 - 144*n^2 + 132*n - 36)*x + (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (24*n^2 - 48*n + 18)*x)*P[n - 1, x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (48*n^3 - 144*n^2 + 132*n - 36)*x + (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (24*n^2 - 48*n + 18)*x)*v[i - 1])/(2*n - 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-2b477f2aa77e3efc","steps":["basis A013610","D+1"],"edges":["e-7e513916b57d5183","e-77739a8efa123beb"],"text":"A109692 \u2192 basis A013610 \u2192 D+1"},"headline":"A109692 \u2192 basis A013610 \u2192 D+1","chain":null}