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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-c68f2dbf6e81cabc"]},{"id":179795,"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-c68f2dbf6e81cabc"]},{"id":179796,"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 real-rootedness)","region_exact":null,"edges":["e-c68f2dbf6e81cabc"]}],"region_exact":{},"incoming_edges":[{"id":"e-c68f2dbf6e81cabc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2124eaaf0381b900","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"e2339bc4437764d2ddc3546ca5a3c5f9af05be335ce22975046eaaa5eade68f3","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-14e332ad5b6d2ede","role":"sequence","name":"Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.","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: 1,  12,    9;\n  2: 1,  28,  117,   112,     25;\n  3: 1,  44,  369,  1008,   1025,     396,       49;\n  4: 1,  60,  765,  3696,   8025,    8316,     4165,      960,        81;\n  5: 1,  76, 1305,  9200,  32225,   60588,    63161,    36800,     11745,      1900,       121;\n  6: 1,  92, 1989, 18544,  91225,  258588,   440461,   459712,    295569,    115900,     26741,      3312,      169;\n  7: 1, 108, 2817, 32752, 209025,  805068,  1963969,  3112896,   3246561,   2236300,   1011681,    294768,    52897,     5292,     225;\n  8: 1, 124, 3789, 52848, 416025, 2041020,  6578005, 14345152,  21524049,  22414300,  16243645,   8164080,  2812329,   647388,   94725,    7936,    289;\n  9: 1, 140, 4905, 79856, 749025, 4482540, 18095945, 50909120, 101773665, 146256300, 152032265, 114545520, 62412545, 24404940, 6741225, 1277696, 157505, 11340, 361;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-8*n+4+(48*n^3-120*n^2+88*n-17)*x+(64*n^4-112*n^3-32*n^2+120*n-35)*x^2+(64*n^4-176*n^3+152*n^2-24*n-11)*x^3-(16*n^3-28*n^2+16*n-3)*x^4)*P(n,x) = -(4*n^2*x^2+(48*n^3+24*n^2-8*n-1)*x^3+(64*n^4+144*n^3+16*n^2-24*n+5)*x^4+(64*n^4+80*n^3+8*n^2+8*n+5)*x^5-(16*n^3+20*n^2+8*n+1)*x^6)*P(n-2,x) + (4*n^2-8*n+4+(48*n^3-56*n^2-40*n+47)*x+(64*n^4+208*n^3-492*n^2-40*n+253)*x^2+(320*n^4+64*n^3-1072*n^2+240*n+504)*x^3+(320*n^4-192*n^3-660*n^2+360*n+276)*x^4+(64*n^4-112*n^3-88*n^2+120*n+49)*x^5-(16*n^3+4*n^2-8*n-3)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*P[n - 2, x] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*P[n - 1, x])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*v[i - 2] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*v[i - 1])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*x^4)); 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-cdb0db4ad473ee28","steps":["basis A008288","(\u03b8+1)\u00b2"],"edges":["e-054f97e0823e3ab6","e-c68f2dbf6e81cabc"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 (\u03b8+1)\u00b2","chain":null}