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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":"7eb724192310bce06bf03729ad118f02ad5fd960ee592b6062392197d04f9ed5","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-f96c16e0c5fd8334","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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:             1159,              1680,               441;\n  1:            29602,             73017,             52920,               9261;\n  2:           706477,           2486568,           3066714,            1481760,             194481;\n  3:         16159804,          74180085,         130544820,          107334990,           38896200,            4084101;\n  4:        359034355,        2036135304,        4673345355,         5482882440,         3381052185,          980184240,           85766121;\n  5:       7809616246,       52778050185,      149655944844,       228993922395,       201495929670,        99402934239,        24014513880,         1801088541;\n  6:     167154015193,     1312015529328,     4433356215540,      8380732911264,      9617744740590,      6770263236912,      2783282158692,       576348333120,        37822859361;\n  7:    3532581112528,    31592108871477,   123985467521496,    279301441579020,    395989630057224,    363550751194302,    213263291962728,     75148618284684,     13616229369960,      794280046581;\n  8:   73895482791391,   741842033630880,  3317171431505085,   8678982726504720,  14663325682898550,  16631564462403408,  12724276291800570,   6397898758881840,   1972651229972955,   317712018632400,   16679880978201;\n  9: 1532766564450250, 17069856524811321, 85682754884366640, 255422200225891545, 501211252455647580, 677445646549913010, 640315231802531208, 419901117629418810, 184739326662713130, 50631381569305845, 7339147630408440, 350277500542221;\n","recurrence_sentence":"The row polynomials satisfy (21*n+40+21*x)*P(n,x) = (399*n+1159+(441*n+1680)*x+441*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1159 + 1680*x + 441*x^2.","mathematica":"P[0, x_] := 1159 + 1680*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*P[n - 1, x])/(21*n + 40 + 21*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1159 + 1680*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*v[i - 1])/(21*n + 40 + 21*x)); 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-5cf6acfdac607d6d","steps":["basis A013610","D+1"],"edges":["e-065a527a7a0e90c1","e-55c5b1d7ba0a603f"],"text":"A038333 \u2192 basis A013610 \u2192 D+1"},"headline":"A038333 \u2192 basis A013610 \u2192 D+1","chain":null}