{"id":"n-968b28a80a14d56e","revision":1,"first_index":0,"fingerprint":"f6-3284531df95da94f","name":"theta_plus_one_squared of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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","8"],["1","20","45"],["1","36","189","224"],["1","56","504","1344","1050"],["1","80","1080","4800","8250","4752"],["1","108","2025","13200","37125","46332","21021"],["1","140","3465","30800","125125","252252","245245","91520"],["1","176","5544","64064","350350","1009008","1569568","1244672","393822"],["1","216","8424","122304","859950","3302208","7277088","9165312","6122142","1679600"],["1","260","12285","218400","1911000","9356256","27289080","48372480","51017850","29393000","7113106"],["1","308","17325","369600","3927000","23750496","87744888","204652800","302182650","273581000","138431986","29953728"],["1","360","23760","598400","7573500","55256256","250699680","736750080","1424575350","1797818000","1423871856","641865600","125550100"],["1","416","31824","933504","13856700","119721888","651819168","2341228032","5679307062","9348653600","10283518960","7231685760","2937872340","524190240"],["1","476","41769","1410864","24249225","244432188","1568439873","6731030592","19877574717","40900359500","58487514085","56949525360","35988936165","13301327340","2181340125"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","66","450","2955","18963","119812","748548","4637205","28537245","174683718","1064611782","6464582943","39132819495","236256182280"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["-2","0"],["-1","2"],["0","3"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-8","4"],["-2","-12","14"],["-1","-2","18"],["0","3","10"],["0","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","2"],["0","-2","4","6"],["0","0","6","6"],["0","0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164923,"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-5812fcfa63c69609"]},{"id":164924,"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-5812fcfa63c69609"]},{"id":164925,"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-5812fcfa63c69609"]},{"id":164926,"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-5812fcfa63c69609"]}],"region_exact":{},"incoming_edges":[{"id":"e-5812fcfa63c69609","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-968b28a80a14d56e","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":"595bfdb6c49348516804ead8fc02317f7b39e49c74b20656397f7ce0b2c6f7de","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-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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-6af8a60dfd64621c","kind":"oeis_match","node_a_id":"n-968b28a80a14d56e","node_b_id":null,"fingerprint":null,"oeis_id":"A002695","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002695","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A002695","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-968b28a80a14d56e","oeis_id":"A002695","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":16428}],"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,   8;\n  2: 1,  20,   45;\n  3: 1,  36,  189,    224;\n  4: 1,  56,  504,   1344,   1050;\n  5: 1,  80, 1080,   4800,   8250,    4752;\n  6: 1, 108, 2025,  13200,  37125,   46332,   21021;\n  7: 1, 140, 3465,  30800, 125125,  252252,  245245,   91520;\n  8: 1, 176, 5544,  64064, 350350, 1009008, 1569568, 1244672,  393822;\n  9: 1, 216, 8424, 122304, 859950, 3302208, 7277088, 9165312, 6122142, 1679600;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+2*n-(n^4+3*n^3+2*n^2)*x)*P(n,x) = -(n^2+2*n-(n^4+3*n^3-2*n^2-12*n-8)*x-(2*n^4+10*n^3+18*n^2+14*n+4)*x^2)*P(n-1,x) - ((2*n+2)*x-(2*n^3+6*n^2+4*n)*x^2-(2*n^3+6*n^2+6*n+2)*x^3)*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[(-(n^2 + 2*n - (n^4 + 3*n^3 - 2*n^2 - 12*n - 8)*x - (2*n^4 + 10*n^3 + 18*n^2 + 14*n + 4)*x^2)*P[n - 1, x] - ((2*n + 2)*x - (2*n^3 + 6*n^2 + 4*n)*x^2 - (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*D[P[n - 1, x], x])/(-n^2 - 2*n + (n^4 + 3*n^3 + 2*n^2)*x)]];\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] = (-(n^2 + 2*n - (n^4 + 3*n^3 - 2*n^2 - 12*n - 8)*x - (2*n^4 + 10*n^3 + 18*n^2 + 14*n + 4)*x^2)*v[i - 1] - ((2*n + 2)*x - (2*n^3 + 6*n^2 + 4*n)*x^2 - (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*deriv(v[i - 1], x))/(-n^2 - 2*n + (n^4 + 3*n^3 + 2*n^2)*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-66e2c3448245b0b7","steps":["(\u03b8+1)\u00b2"],"edges":["e-5812fcfa63c69609"],"text":"A033282 \u2192 (\u03b8+1)\u00b2"},"headline":"A033282 \u2192 (\u03b8+1)\u00b2","chain":null}
