{"id":"n-5f6cf92c5fbb97a3","revision":0,"first_index":0,"fingerprint":"f6-0bf5df792ca502a7","name":"shift_x_plus_one of gamma_polynomial of Pascal-(1,7,1) array.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["8","7"],["15","14"],["71","119","49"],["176","322","147"],["673","1652","1323","343"],["1905","5138","4606","1372"],["6616","21413","25431","13034","2401"],["19951","70714","93639","54880","12005"],["66263","266917","421547","324135","120050","16807"],["205920","901572","1572018","1363768","588245","100842"],["669761","3233832","6391266","6583542","3697540","1058841","117649"],["2111201","10986276","23706396","27134044","17361631","5882450","823543"],["6799528","38311427","91082082","117957700","89329205","39177117","9058973","823543"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","15","29","239","645","3991","13021","68895","251189","1215719","4732365","21752431","88005541","392539575"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-14","-14"],["15","14"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-29","-57","-28"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-aca1e71c58a7aa9f","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","7"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174327,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-cab99da7be737389"]},{"id":174328,"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-6d848c1987f20f0f","e-cab99da7be737389"]},{"id":174329,"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-6d848c1987f20f0f","e-cab99da7be737389"]},{"id":174330,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-6d848c1987f20f0f"]},{"id":174331,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-6d848c1987f20f0f"]}],"region_exact":{},"incoming_edges":[{"id":"e-6d848c1987f20f0f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5f6cf92c5fbb97a3","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d6363c46208a444d91e9c9a73f401c2bc7cd8dfffc593aff1d36d4531f233a39","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-695d6cb541611b1e","role":"sequence","name":"gamma_polynomial of Pascal-(1,7,1) array.","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]"}]},{"id":"e-aca1e71c58a7aa9f","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-5f6cf92c5fbb97a3","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"d7e86bb548113a27c7b3d34d6432b424eae79e01349c501598571e30ac830cfa","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-cab99da7be737389","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-5f6cf92c5fbb97a3","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"49198af6aca8944dc66fe5f2cd98f4680a8d69f62d2029d279b3680b27f9d91e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-695d6cb541611b1e","role":"sequence","name":"gamma_polynomial of Pascal-(1,7,1) array.","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;\n  2:     8,     7;\n  3:    15,    14;\n  4:    71,   119,    49;\n  5:   176,   322,   147;\n  6:   673,  1652,  1323,   343;\n  7:  1905,  5138,  4606,  1372;\n  8:  6616, 21413, 25431, 13034,  2401;\n  9: 19951, 70714, 93639, 54880, 12005;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (15*n-14+(14*n-14)*x)*P(n-1,x) - (29+57*x+28*x^2)*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[((15*n - 14 + (14*n - 14)*x)*P[n - 1, x] - (29 + 57*x + 28*x^2)*D[P[n - 1, x], x])/(n)]];\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] = ((15*n - 14 + (14*n - 14)*x)*v[i - 1] - (29 + 57*x + 28*x^2)*deriv(v[i - 1], x))/(n)); 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-1e2f1b7f566b371c","steps":["\u03b3","basis A007318"],"edges":["e-c807fb938e5557d1","e-6d848c1987f20f0f"],"text":"A081582 \u2192 \u03b3 \u2192 basis A007318"},"headline":"A081582 \u2192 \u03b3 \u2192 basis A007318","chain":null}