{"id":"n-2a2ed4d5f9d376c7","revision":0,"first_index":0,"fingerprint":"f6-ea3a19fb0ca3a7eb","name":"theta_plus_one_squared of Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, 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","4"],["1","12","18"],["1","28","108","96"],["1","60","450","960","600"],["1","124","1620","6240","9000","4320"],["1","252","5418","33600","84000","90720","35280"],["1","508","17388","163296","630000","1149120","987840","322560"],["1","1020","54450","745920","4170600","11430720","16299360","11612160","3265920"],["1","2044","167940","3274080","25515000","98612640","207446400","241920000","146966400","36288000"],["1","4092","513018","13992000","148038000","775383840","2257461360","3832012800","3772137600","1995840000","439084800"],["1","8188","1557468","58704096","827640000","5718176640","22134530880","51295749120","72748368000","61871040000","28979596800","5748019200"],["1","16380","4709250","243122880","4505100600","40268067840","201640158720","612738846720","1174104771840","1427025600000","1067415148800","448345497600","80951270400"],["1","32764","14201460","997607520","24045021000","274045131360","1740336998400","6745477939200","16770923769600","27315864576000","29008576396800","19353580646400","7366565606400","1220496076800"],["1","65532","42751818","4066171200","126460152000","1817394939360","14420394228240","69875476070400","219236278060800","459502243200000","649616301734400","611991604224000","368328280320000","128152088064000","19615115520000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","31","233","2071","21305","249271","3270713","47580151","760192505","13234467511","249383390393","5057242311031","109820924003705","2542685745501751"],"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"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179465,"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-f07788e7e2379024"]},{"id":179466,"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-f07788e7e2379024"]},{"id":179467,"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-f07788e7e2379024"]}],"region_exact":{},"incoming_edges":[{"id":"e-f07788e7e2379024","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-2a2ed4d5f9d376c7","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":"07f32c2cbb80aa78b594138921e984d3a29cd435564d02b7c1e3a23f41378691","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-677d34c7af372519","role":"sequence","name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, 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-21ab68f12ed6d1e7","kind":"fingerprint_related","node_a_id":"n-2a2ed4d5f9d376c7","node_b_id":"n-b9e1bcc0efba62aa","fingerprint":"f6-ea3a19fb0ca3a7eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea3a19fb0ca3a7eb"},"other_id":"n-b9e1bcc0efba62aa","other":{"id":"n-b9e1bcc0efba62aa","name":"theta_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","first_index":1,"fingerprint":"f6-ea3a19fb0ca3a7eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","18"]],"card_rows":[["1"],["1","4"],["1","12","18"],["1","28","108","96"]],"row_sums_preview":["1","5","31","233","2071","21305","249271","3270713"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d938c1937a93c4df","steps":["\u03b8+1"],"edges":["e-55ac75c563f7d5fb"],"text":"A019538 \u2192 \u03b8+1"},"headline":"A019538 \u2192 \u03b8+1","chain":null}},{"id":"l-cc7bd97c42756437","kind":"fingerprint_related","node_a_id":"n-2b66f014abf34c41","node_b_id":"n-2a2ed4d5f9d376c7","fingerprint":"f6-ea3a19fb0ca3a7eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea3a19fb0ca3a7eb"},"other_id":"n-2b66f014abf34c41","other":{"id":"n-2b66f014abf34c41","name":"derivative of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","first_index":1,"fingerprint":"f6-ea3a19fb0ca3a7eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","4"]],"card_rows":[["1"],["1","4"],["1","12","18"],["1","28","108","96"]],"row_sums_preview":["0","1","5","31","233","2071","21305","249271"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2cabfd060b04f15d","steps":["D"],"edges":["e-196070c9ee190f69"],"text":"A131689 \u2192 D"},"headline":"A131689 \u2192 D","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,    4;\n  2: 1,   12,     18;\n  3: 1,   28,    108,      96;\n  4: 1,   60,    450,     960,      600;\n  5: 1,  124,   1620,    6240,     9000,     4320;\n  6: 1,  252,   5418,   33600,    84000,    90720,     35280;\n  7: 1,  508,  17388,  163296,   630000,  1149120,    987840,    322560;\n  8: 1, 1020,  54450,  745920,  4170600, 11430720,  16299360,  11612160,   3265920;\n  9: 1, 2044, 167940, 3274080, 25515000, 98612640, 207446400, 241920000, 146966400, 36288000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*x^2*P(n-2,x) - (x^2+x^3)*d/dx P(n-2,x) + (1+4*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-2*x^2*P[n - 2, x] - (x^2 + x^3)*D[P[n - 2, x], x] + (1 + 4*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -2*x^2*v[i - 2] - (x^2 + x^3)*deriv(v[i - 2], x) + (1 + 4*x)*v[i - 1] + (x + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-677d34c7af372519","steps":["(\u03b8+1)\u00b2"],"edges":["e-f07788e7e2379024"],"text":"A028246 \u2192 (\u03b8+1)\u00b2"},"headline":"A028246 \u2192 (\u03b8+1)\u00b2","chain":null}