{"id":"n-0abf36e1d724d74f","revision":1,"first_index":1,"fingerprint":"f6-915ca4b22aed877c","name":"theta_plus_one_squared of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","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"],["1","4"],["1","12","9"],["1","28","54","16"],["1","60","225","160","25"],["1","124","810","1040","375","36"],["1","252","2709","5600","3500","756","49"],["1","508","8694","27216","26250","9576","1372","64"],["1","1020","27225","124320","173775","95256","22638","2304","81"],["1","2044","83970","545680","1063125","821772","288120","48000","3645","100"],["1","4092","256509","2332000","6168250","6461532","3135363","760320","93555","5500","121"],["1","8188","778734","9784016","34485000","47651472","30742404","10177728","1804275","170500","7986","144"],["1","16380","2354625","40520480","187712525","335567232","280055776","121575168","29119662","3932500","294151","11232","169"],["1","32764","7100730","166267920","1001875875","2283709428","2417134720","1338388480","415945530","75275200","7993986","484848","15379","196"],["1","65532","21375909","677695200","5269173000","15144957828","20028325317","13864181760","5437407690","1266265000","179016838","15331680","768950","20580","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","22","99","471","2386","12867","73681","446620","2856457","19217243","135610448","1001159901","7714225057","61904585510"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":"e-6385ba87ef72ed57","recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","4"],["1","12","9"]],"recurrence":{"denominator":{"coeffs":[["4"]]},"terms":[{"coeff":{"coeffs":[["0","6","-18","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","3","-9"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","-4"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-6","12","-12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164097,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-72cd2be7747bf330","e-b4b600b457a94a2b"]},{"id":164098,"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-72cd2be7747bf330","e-b4b600b457a94a2b"]},{"id":164099,"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-72cd2be7747bf330","e-b4b600b457a94a2b"]},{"id":164100,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-72cd2be7747bf330"]}],"region_exact":{},"incoming_edges":[{"id":"e-6385ba87ef72ed57","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-0abf36e1d724d74f","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":"ea271f4fb1c5a3cc0537009e96a1366ef1b8f9c01112dceece3fc8fa541e52d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-72cd2be7747bf330","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-0abf36e1d724d74f","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":"8ac33e6db4f29fcfb0bed017329909a4eab7eeabb0e8e562280fbbf9c28ab668","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3cb22164e2f83fe1","role":"sequence","name":"Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","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]"}]},{"id":"e-b4b600b457a94a2b","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-0abf36e1d724d74f","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":"0f36cbabdba687ef97a3bc493e655f1fac5d82ccfea5f07c4574355fad9a5944","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f25b32370a8d627e","role":"sequence","name":"Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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-38124c1a330d4896","kind":"oeis_match","node_a_id":"n-0abf36e1d724d74f","node_b_id":null,"fingerprint":null,"oeis_id":"A033452","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A033452","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A033452","relation":"match_row_sums"}],"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: 1,    4;\n   3: 1,   12,     9;\n   4: 1,   28,    54,     16;\n   5: 1,   60,   225,    160,      25;\n   6: 1,  124,   810,   1040,     375,     36;\n   7: 1,  252,  2709,   5600,    3500,    756,     49;\n   8: 1,  508,  8694,  27216,   26250,   9576,   1372,    64;\n   9: 1, 1020, 27225, 124320,  173775,  95256,  22638,  2304,   81;\n  10: 1, 2044, 83970, 545680, 1063125, 821772, 288120, 48000, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-2,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (1+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 2, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 2] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); 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-3cb22164e2f83fe1","steps":["(\u03b8+1)\u00b2"],"edges":["e-72cd2be7747bf330"],"text":"A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A008277 \u2192 (\u03b8+1)\u00b2","chain":null}
