{"id":"n-3c6f163a8ebc0747","revision":0,"first_index":0,"fingerprint":"f6-ec0cbe08708f3ce3","name":"mobius_x_over_x_plus_one of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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","2"],["1","9","9"],["1","22","60","40"],["1","43","294","502","251"],["1","74","1047","3468","4155","1662"],["1","117","2999","19379","43727","40845","13615"],["1","174","7358","87032","330810","547564","415240","118640"],["1","247","16084","322984","2093712","5689710","7514980","4815780","1203945"],["1","338","32153","1025500","11140390","47065072","97069686","105535296","58350825","12966850"],["1","449","59861","2869885","50149802","332765716","1039592828","1738985878","1613474257","785818525","157163705"],["1","582","105168","7245928","194442115","2026796910","9330039782","22699824832","31632698655","25481848890","11071416180","2012984760"],["1","739","176082","16804314","663044601","10633732625","72735538892","254075478716","507712047749","608378567125","433925495170","170318062770","28386343795"],["1","922","283083","36297228","2027201605","48465357610","496047267667","2455133997752","6813268500315","11457989413770","11988415735065","7654310030780","2737299039475","421122929150"],["1","1133","439587","73818983","5648698825","194608184055","2954004423121","20871406280139","80027039175891","184298119445585","268167333071271","249472690520565","144332822710495","47424941980185","6774991711455"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","19","123","1091","10407","120683","1506819","21657443","333186111","5720880907","104457403803","2086845292579","44074116054423","1004523680461291"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","-1","-2","-1"],["0","1","2","1"],["0","1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181267,"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-9e5843cdcdfb6273"]},{"id":181268,"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-9e5843cdcdfb6273"]},{"id":181269,"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-9e5843cdcdfb6273"]}],"region_exact":{},"incoming_edges":[{"id":"e-9e5843cdcdfb6273","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-3c6f163a8ebc0747","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":"756b54e0797a91e4494c017a914d354f3a6556930a3bbd5748df053d76daa0d8","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-74f9a0f107d622a7","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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-2f08b7f39a9c8c1a","kind":"fingerprint_related","node_a_id":"n-8d5da6cb1d1876fb","node_b_id":"n-3c6f163a8ebc0747","fingerprint":"f6-ec0cbe08708f3ce3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ec0cbe08708f3ce3"},"other_id":"n-8d5da6cb1d1876fb","other":{"id":"n-8d5da6cb1d1876fb","name":"basis_map_A007318 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","first_index":0,"fingerprint":"f6-ec0cbe08708f3ce3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["9","9","1"]],"card_rows":[["1"],["2","1"],["9","9","1"],["40","60","22","1"]],"row_sums_preview":["1","3","19","123","1091","10407","120683","1506819"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74f9a0f107d622a7","steps":["x \u21a6 x+1"],"edges":["e-51c9e381d6371b4c"],"text":"A154233 \u2192 x \u21a6 x+1"},"headline":"A154233 \u2192 x \u21a6 x+1","chain":null}}],"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,   2;\n  2: 1,   9,     9;\n  3: 1,  22,    60,      40;\n  4: 1,  43,   294,     502,      251;\n  5: 1,  74,  1047,    3468,     4155,     1662;\n  6: 1, 117,  2999,   19379,    43727,    40845,    13615;\n  7: 1, 174,  7358,   87032,   330810,   547564,   415240,    118640;\n  8: 1, 247, 16084,  322984,  2093712,  5689710,  7514980,   4815780,  1203945;\n  9: 1, 338, 32153, 1025500, 11140390, 47065072, 97069686, 105535296, 58350825, 12966850;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = ((n^2+n-1)*x+(2*n^2+2*n-2)*x^2+(n^2+n-1)*x^3)*P(n-2,x) + (1+3*x+2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^2 + n - 1)*x + (2*n^2 + 2*n - 2)*x^2 + (n^2 + n - 1)*x^3)*P[n - 2, x] + (1 + 3*x + 2*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + n - 1)*x + (2*n^2 + 2*n - 2)*x^2 + (n^2 + n - 1)*x^3)*v[i - 2] + (1 + 3*x + 2*x^2)*v[i - 1])/(1 + x)); 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-74f9a0f107d622a7","steps":["x \u21a6 x/(x+1)"],"edges":["e-9e5843cdcdfb6273"],"text":"A154233 \u2192 x \u21a6 x/(x+1)"},"headline":"A154233 \u2192 x \u21a6 x/(x+1)","chain":null}