{"id":"n-9ad63aa932f548eb","revision":0,"first_index":0,"fingerprint":"f6-c3f75468b073d346","name":"theta_plus_one of reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","8","12"],["1","10","24","16"],["1","12","39","48","20"],["1","14","57","100","80","24"],["1","16","78","176","205","120","28"],["1","18","102","280","425","366","168","32"],["1","20","129","416","775","876","595","224","36"],["1","22","159","588","1295","1806","1617","904","288","40"],["1","24","192","800","2030","3360","3724","2752","1305","360","44"],["1","26","228","1056","3030","5796","7644","7008","4401","1810","440","48"],["1","28","267","1360","4350","9432","14406","15744","12285","6700","2431","528","52"],["1","30","309","1716","6050","14652","25410","32208","29997","20350","9801","3180","624","56"],["1","32","354","2128","8195","21912","42504","61248","66231","53680","32186","13872","4069","728","60"],["1","34","402","2600","10855","31746","68068","109824","135135","127270","91234","48984","19097","5110","840","64"],["1","36","453","3136","14105","44772","105105","187616","258687","277420","231231","148512","72163","25676","6315","960","68"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["21","51","120","276","624","1392","3072","6720","14592","31488","67584","144384","307200","651264","1376256"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","8","12"]],"recurrence":{"denominator":{"coeffs":[["1","6","4"],["0","1","2"]]},"terms":[{"coeff":{"coeffs":[["1","8","13","6"],["0","1","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172853,"position":0,"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-c88f5390bdb08588"]},{"id":172854,"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-c88f5390bdb08588"]},{"id":172855,"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-c88f5390bdb08588"]},{"id":172856,"position":3,"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-c88f5390bdb08588"]}],"region_exact":{},"incoming_edges":[{"id":"e-c88f5390bdb08588","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-9ad63aa932f548eb","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"f6231bb8dc1bb5be8300de1f832288506f4718432400bf1652394e362b16fcd7","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-3a033458c346939f","role":"sequence","name":"reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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-80afdd5e593df148","kind":"fingerprint_related","node_a_id":"n-9ad63aa932f548eb","node_b_id":"n-6174136b207ae39c","fingerprint":"f6-c3f75468b073d346","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c3f75468b073d346"},"other_id":"n-6174136b207ae39c","other":{"id":"n-6174136b207ae39c","name":"theta_plus_one of basis_map_A013609 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-c3f75468b073d346","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","8","12"],["2","20","48","32"],["4","48","156","192","80"]],"card_rows":[["1","8","12"],["2","20","48","32"],["4","48","156","192","80"],["8","112","456","800","640","192"]],"row_sums_preview":["21","102","480","2208","9984","44544","196608","860160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["basis A013609","\u03b8+1"],"edges":["e-4ff089327fda7415","e-11fa58ed434502c7"],"text":"A159854 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A159854 \u2192 basis A013609 \u2192 \u03b8+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,  8,  12;\n  1: 1, 10,  24,   16;\n  2: 1, 12,  39,   48,   20;\n  3: 1, 14,  57,  100,   80,   24;\n  4: 1, 16,  78,  176,  205,  120,   28;\n  5: 1, 18, 102,  280,  425,  366,  168,   32;\n  6: 1, 20, 129,  416,  775,  876,  595,  224,   36;\n  7: 1, 22, 159,  588, 1295, 1806, 1617,  904,  288,   40;\n  8: 1, 24, 192,  800, 2030, 3360, 3724, 2752, 1305,  360,  44;\n  9: 1, 26, 228, 1056, 3030, 5796, 7644, 7008, 4401, 1810, 440, 48;\n","recurrence_sentence":"The row polynomials satisfy (1+(n+6)*x+(2*n+4)*x^2)*P(n,x) = (1+(n+8)*x+(3*n+13)*x^2+(2*n+6)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 12*x^2.","mathematica":"P[0, x_] := 1 + 8*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 8)*x + (3*n + 13)*x^2 + (2*n + 6)*x^3)*P[n - 1, x])/(1 + (n + 6)*x + (2*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 8)*x + (3*n + 13)*x^2 + (2*n + 6)*x^3)*v[i - 1])/(1 + (n + 6)*x + (2*n + 4)*x^2)); 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-f48bed27dbc29c5f","steps":["reverse","\u03b8+1"],"edges":["e-6248136e99299279","e-c88f5390bdb08588"],"text":"A391533 \u2192 reverse \u2192 \u03b8+1"},"headline":"A391533 \u2192 reverse \u2192 \u03b8+1","chain":null}