{"id":"n-dccb22bb1a700e42","revision":0,"first_index":1,"fingerprint":"f6-ab299508b34125dc","name":"mobius_x_over_x_plus_one of Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","7","7"],["1","24","57","35"],["1","89","366","491","214"],["1","346","2194","4841","4514","1523"],["1","1371","12921","42064","63141","44976","12349"],["1","5468","76719","348391","749909","846168","485449","112052"],["1","21853","465928","2848453","8249641","13107541","11782198","5640769","1120849"],["1","87390","2911824","23366337","87464817","183493143","229726398","171079821","70143417","12219767"],["1","349535","18722331","193712298","912329958","2427831405","3970295022","4088828724","2594404800","928973141","143942992"],["1","1398112","123366517","1628128239","9473830530","31128024390","63869328984","85299121029","74460181872","41099885432","13049105618","1819256321"],["1","5592417","828612786","13895375207","98663233824","392656498794","983438785176","1635622868538","1841754943359","1393030431572","679678002246","193679465244","24526654381"],["1","22369634","5643701790","120539721017","1035437652820","4920696511875","14746793311392","29704794203514","41426958399111","40255078262972","26829567380758","11719455992544","3027161086429","350974470746"],["1","89478499","38812453837","1063499358596","10984192628267","61654263072830","217787380707183","520661962880916","874515271456113","1048152438090185","894648543287357","532480614348266","210403077554827","49672880899591","5308470041299"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","117","1161","13419","176823","2624157","43237233","780492915","15279390207","321951627045","7257780463545","174143123064603","4427371496257767"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","9","14","6"],["0","-3","-6","-3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","9","9","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179630,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-cc32fec68ed6505c"]},{"id":179631,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-cc32fec68ed6505c"]},{"id":179632,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-cc32fec68ed6505c"]}],"region_exact":{},"incoming_edges":[{"id":"e-cc32fec68ed6505c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-dccb22bb1a700e42","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":"76249e30ea870fbc20a17bb577f507d3bf178bfb2e6807d471b83aef752166b3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     2;\n   3: 1,     7,       7;\n   4: 1,    24,      57,       35;\n   5: 1,    89,     366,      491,      214;\n   6: 1,   346,    2194,     4841,     4514,      1523;\n   7: 1,  1371,   12921,    42064,    63141,     44976,     12349;\n   8: 1,  5468,   76719,   348391,   749909,    846168,    485449,    112052;\n   9: 1, 21853,  465928,  2848453,  8249641,  13107541,  11782198,   5640769,  1120849;\n  10: 1, 87390, 2911824, 23366337, 87464817, 183493143, 229726398, 171079821, 70143417, 12219767;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (1-(3*n-9)*x-(6*n-14)*x^2-(3*n-6)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 - (3*n - 9)*x - (6*n - 14)*x^2 - (3*n - 6)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 - (3*n - 9)*x - (6*n - 14)*x^2 - (3*n - 6)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cab14672969f6507","steps":["x \u21a6 x/(x+1)"],"edges":["e-cc32fec68ed6505c"],"text":"A111577 \u2192 x \u21a6 x/(x+1)"},"headline":"A111577 \u2192 x \u21a6 x/(x+1)","chain":null}
