{"id":"n-864f996c2a017221","revision":0,"first_index":0,"fingerprint":"f6-8f228763548ec508","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","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"],["2","2"],["9","15","6"],["44","104","80","20"],["225","690","780","385","70"],["1182","4458","6636","4872","1764","252"],["6321","28301","52220","50820","27510","7854","924"],["34232","177440","390528","473040","340560","145728","34320","3432"],["187137","1102212","2817360","4081770","3665970","2090088","738738","148005","12870"],["1030490","6797570","19787240","33359480","35895860","25565540","12052040","3626480","632060","48620"],["5707449","41682267","136113714","261712308","328113786","280269990","165189024","66337128","17371926","2678962","184756"],["31760676","254399976","920902320","1988581140","2846156040","2834971776","2005357536","1007388720","352212120","81628560","11286912","705432"],["177435297","1546656150","6146648508","14726546835","23690006340","26956561632","22248025800","13419382548","5871083790","1817075260","377641264","47322730","2704156"],["994551222","9372259722","40565950740","106778936880","190704701880","244035309672","230194833792","162065405040","85161785940","32989391820","9157104264","1724835504","197611400","10400600"],["5590402785","56633660505","265179315960","760667352120","1493356833540","2122610876100","2252583950760","1813171365720","1112411718990","517671069630","179876594040","45255982200","7793763900","822390300","40116600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","30","248","2150","19164","173950","1599280","14844150","138795380","1305361310","12335351208","117027090310","1113953078476","10633665393150"],"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":[["-1","0"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"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-f4e0dab53593f1d4","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["2"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"],["-9","-6"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176911,"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-e077029e7893d049"]},{"id":176912,"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-e077029e7893d049","e-e2f2ebdd80d8e4a2"]},{"id":176913,"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-e077029e7893d049","e-e2f2ebdd80d8e4a2"]},{"id":176914,"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-e2f2ebdd80d8e4a2"]},{"id":176915,"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-e2f2ebdd80d8e4a2"]}],"region_exact":{},"incoming_edges":[{"id":"e-e2f2ebdd80d8e4a2","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-864f996c2a017221","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":"b33d5dbb385f1262e01eed8acf0615d2504b033640bb3c7e6e489ba7f701a1ee","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-2fa133578d3075fe","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","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-f4e0dab53593f1d4","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-864f996c2a017221","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":"9273f51040ff1059fb79e474168c52c270da91616fd2fce97109b8ea7117c8ed","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-e077029e7893d049","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-864f996c2a017221","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":"1b9888beec6f954c6d31352c35c13dff2178070df4a56c617fe98386231ba27b","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-2fa133578d3075fe","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","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-c3f8d0912d378542","kind":"fingerprint_related","node_a_id":"n-864f996c2a017221","node_b_id":"n-79beb834849e7646","fingerprint":"f6-8f228763548ec508","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8f228763548ec508"},"other_id":"n-79beb834849e7646","other":{"id":"n-79beb834849e7646","name":"shift_x_plus_one of Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.","first_index":0,"fingerprint":"f6-8f228763548ec508","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["18","30","12"]],"card_rows":[["1"],["2","2"],["18","30","12"],["264","624","480","120"]],"row_sums_preview":["1","4","60","1488","51600","2299680","125244000","8060371200"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4cba07695b3e3f90","steps":["x \u21a6 x+1"],"edges":["e-0eaee4ae3e4370a6"],"text":"A357367 \u2192 x \u21a6 x+1"},"headline":"A357367 \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:       2,       2;\n  2:       9,      15,        6;\n  3:      44,     104,       80,       20;\n  4:     225,     690,      780,      385,       70;\n  5:    1182,    4458,     6636,     4872,     1764,      252;\n  6:    6321,   28301,    52220,    50820,    27510,     7854,      924;\n  7:   34232,  177440,   390528,   473040,   340560,   145728,    34320,    3432;\n  8:  187137, 1102212,  2817360,  4081770,  3665970,  2090088,   738738,  148005,  12870;\n  9: 1030490, 6797570, 19787240, 33359480, 35895860, 25565540, 12052040, 3626480, 632060, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n-1+2*n*x)*P(n-1,x) + (4+6*x+2*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[((3*n - 1 + 2*n*x)*P[n - 1, x] + (4 + 6*x + 2*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] = ((3*n - 1 + 2*n*x)*v[i - 1] + (4 + 6*x + 2*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-2fa133578d3075fe","steps":["x \u21a6 x+1"],"edges":["e-e077029e7893d049"],"text":"A386789 \u2192 x \u21a6 x+1"},"headline":"A386789 \u2192 x \u21a6 x+1","chain":null}