{"id":"n-ab6ae8e31e413a35","revision":0,"first_index":1,"fingerprint":"f6-261a981b55e1ac6d","name":"shift_x_plus_one of reversal of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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"],["2","1"],["6","7","2"],["16","28","16","3"],["44","104","91","35","5"],["120","356","418","243","70","8"],["328","1172","1730","1351","589","136","13"],["896","3744","6656","6528","3816","1330","256","21"],["2448","11712","24360","28776","21121","9866","2865","473","34"],["6688","36048","85872","118680","104890","61489","23914","5951","860","55"],["18272","109552","294096","465576","481390","339739","165766","55223","12023","1545","89"],["49920","329536","984320","1756272","2080048","1717196","1008456","421345","122756","23754","2748","144"],["136384","982912","3233296","6419856","8569972","8103664","5566247","2798629","1022338","264646","46086","4848","233"],["372608","2911040","10456672","22868048","33973032","36208108","28484706","16748759","7361308","2388751","556332","88068","8494","377"],["1017984","8569664","33376096","79715280","130446920","154726348","137190978","92381887","47478185","18533745","5409863","1145068","166152","14795","610"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","63","279","1215","5319","23247","101655","444447","1943271","8496495","37149111","162426303","710173575"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_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":[["-18","-27","-10"],["10","14","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","-28","-21","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172839,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-cf2481f3b4c8561c"]},{"id":172840,"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-cf2481f3b4c8561c"]},{"id":172841,"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-cf2481f3b4c8561c"]}],"region_exact":{},"incoming_edges":[{"id":"e-cf2481f3b4c8561c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ab6ae8e31e413a35","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":"2ed40cd5abc8e147b99b3e2a09a763b4a3f4ee49d9a982c99f671236e390b669","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-305352172c67c5f0","role":"sequence","name":"reversal of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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":[{"id":"l-b338c9bc460a7768","kind":"fingerprint_related","node_a_id":"n-f263ca9517b1fbce","node_b_id":"n-ab6ae8e31e413a35","fingerprint":"f6-261a981b55e1ac6d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-261a981b55e1ac6d"},"other_id":"n-f263ca9517b1fbce","other":{"id":"n-f263ca9517b1fbce","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","first_index":1,"fingerprint":"f6-261a981b55e1ac6d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["2","7","6"]],"card_rows":[["1"],["1","2"],["2","7","6"],["3","16","28","16"]],"row_sums_preview":["1","3","15","63","279","1215","5319","23247"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["x \u21a6 x/(x+1)"],"edges":["e-5274886585caff6b"],"text":"A063967 \u2192 x \u21a6 x/(x+1)"},"headline":"A063967 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:    2,     1;\n   3:    6,     7,     2;\n   4:   16,    28,    16,      3;\n   5:   44,   104,    91,     35,      5;\n   6:  120,   356,   418,    243,     70,     8;\n   7:  328,  1172,  1730,   1351,    589,   136,    13;\n   8:  896,  3744,  6656,   6528,   3816,  1330,   256,   21;\n   9: 2448, 11712, 24360,  28776,  21121,  9866,  2865,  473,  34;\n  10: 6688, 36048, 85872, 118680, 104890, 61489, 23914, 5951, 860, 55;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (10*n-18+(14*n-27)*x+(5*n-10)*x^2)*P(n-1,x) - (12+28*x+21*x^2+5*x^3)*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[((10*n - 18 + (14*n - 27)*x + (5*n - 10)*x^2)*P[n - 1, x] - (12 + 28*x + 21*x^2 + 5*x^3)*D[P[n - 1, x], x])/(n - 1)]];\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] = ((10*n - 18 + (14*n - 27)*x + (5*n - 10)*x^2)*v[i - 1] - (12 + 28*x + 21*x^2 + 5*x^3)*deriv(v[i - 1], x))/(n - 1)); 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-c72653d9ab0ee8be","steps":["reverse","x \u21a6 x+1"],"edges":["e-2c112d9ec33e1fb9","e-cf2481f3b4c8561c"],"text":"A063967 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A063967 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}