{"id":"n-78fa6a967cf79648","revision":1,"first_index":0,"fingerprint":"f6-f80b58a897df2b0a","name":"mobius_x_over_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","3"],["1","7","10"],["1","12","40","37"],["1","18","103","221","151"],["1","25","215","790","1241","674"],["1","33","395","2180","5836","7218","3263"],["1","42","665","5110","20496","42938","43807","17007"],["1","52","1050","10682","59591","186354","319985","278111","94828"],["1","63","1578","20496","151347","656901","1676473","2437098","1847242","562595"],["1","75","2280","36780","346857","1990653","7031515","15123540","19062151","12825759","3535027"],["1","88","3190","62535","732897","5369826","25037881","74297410","137929451","153507134","92962365","23430840"],["1","102","4345","101695","1449492","13199076","78496594","306819414","783200726","1278273480","1274286388","702294568","163254885"],["1","117","5785","159302","2713282","30058743","222083290","1105682292","3708735602","8294775632","12076730172","10908546948","5521005933","1192059223"],["1","133","7553","241696","4847843","64211147","577149001","3566985136","15214459260","44598754200","88683837606","116548030508","96293931189","45093176851","9097183602"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","18","90","494","2946","18926","130066","950654","7353794","59954638","513333618","4601380766","43062556322","419742815726"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","4","4","0","-1"],["0","1","3","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-3","-3","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165053,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-0bdaeb662c106d9c"]},{"id":165054,"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-0bdaeb662c106d9c"]},{"id":165055,"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-0bdaeb662c106d9c"]}],"region_exact":{},"incoming_edges":[{"id":"e-0bdaeb662c106d9c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-78fa6a967cf79648","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"87a213090553867cf91ae3815af3e02264a1210ffadeadf51ef537a423af0d43","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0990b5be9f344750","role":"sequence","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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-be357e2d7c55353e","kind":"fingerprint_related","node_a_id":"n-78fa6a967cf79648","node_b_id":"n-3684821bf6f3ad98","fingerprint":"f6-f80b58a897df2b0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f80b58a897df2b0a"},"other_id":"n-3684821bf6f3ad98","other":{"id":"n-3684821bf6f3ad98","name":"shift_x_plus_one of Triangle read by rows: 2-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-f80b58a897df2b0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["10","7","1"]],"card_rows":[["1"],["3","1"],["10","7","1"],["37","40","12","1"]],"row_sums_preview":["1","4","18","90","494","2946","18926","130066"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["x \u21a6 x+1"],"edges":["e-9842acf899b9e46a"],"text":"A143494 \u2192 x \u21a6 x+1"},"headline":"A143494 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f06c67e6352dd21d","kind":"fingerprint_related","node_a_id":"n-78fa6a967cf79648","node_b_id":"n-39bcc602ce65607f","fingerprint":"f6-f80b58a897df2b0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f80b58a897df2b0a"},"other_id":"n-39bcc602ce65607f","other":{"id":"n-39bcc602ce65607f","name":"one_plus_derivative of Triangle read by rows: A008277 * A007318.","first_index":0,"fingerprint":"f6-f80b58a897df2b0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["10","7","1"]],"card_rows":[["1"],["3","1"],["10","7","1"],["37","40","12","1"]],"row_sums_preview":["1","4","18","90","494","2946","18926","130066"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["basis A049020","D+1"],"edges":["e-3c112fda27f20cb0","e-70027a74f76c6c1d"],"text":"A007318 \u2192 basis A049020 \u2192 D+1"},"headline":"A007318 \u2192 basis A049020 \u2192 D+1","chain":null}},{"id":"l-eb64f34b7ed9213b","kind":"oeis_match","node_a_id":"n-78fa6a967cf79648","node_b_id":null,"fingerprint":null,"oeis_id":"A355247","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A355247","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A355247","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-78fa6a967cf79648","oeis_id":"A355247","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":27511}],"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,  3;\n  2: 1,  7,   10;\n  3: 1, 12,   40,    37;\n  4: 1, 18,  103,   221,    151;\n  5: 1, 25,  215,   790,   1241,    674;\n  6: 1, 33,  395,  2180,   5836,   7218,    3263;\n  7: 1, 42,  665,  5110,  20496,  42938,   43807,   17007;\n  8: 1, 52, 1050, 10682,  59591, 186354,  319985,  278111,   94828;\n  9: 1, 63, 1578, 20496, 151347, 656901, 1676473, 2437098, 1847242, 562595;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(n+4)*x+(3*n+4)*x^2+3*n*x^3+(n-1)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+x^5)*d/dx 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[((1 + (n + 4)*x + (3*n + 4)*x^2 + 3*n*x^3 + (n - 1)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((1 + (n + 4)*x + (3*n + 4)*x^2 + 3*n*x^3 + (n - 1)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-0990b5be9f344750","steps":["x \u21a6 x/(x+1)"],"edges":["e-0bdaeb662c106d9c"],"text":"A126351 \u2192 x \u21a6 x/(x+1)"},"headline":"A126351 \u2192 x \u21a6 x/(x+1)","chain":null}