{"id":"n-8fb868076fadaac9","revision":0,"first_index":1,"fingerprint":"f6-5eb99b00596b7bfc","name":"theta_plus_one of Triangle of Ward numbers T(n,k) read by rows.","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","6"],["1","20","45"],["1","50","315","420"],["1","112","1470","5040","4725"],["1","238","5754","37800","86625","62370"],["1","492","20475","227920","952875","1621620","945945"],["1","1002","68805","1211980","8183175","24594570","33108075","16216200"],["1","2024","222948","5948800","60610550","285044760","662161500","735134400","310134825"],["1","4070","705276","27659632","407156750","2801258460","9955966020","18745927200","17677685025","6547290750"],["1","8164","2195193","123803680","2554401850","24624960360","125250054930","354775861440","559793359125","458310352500","151242416325"],["1","16354","6757023","539118580","15248082850","199859559900","1393874552070","5557922370000","13020597114525","17644948571250","12754777110075","3794809718700"],["1","32736","20639034","2300624144","87696684075","1528515948960","14187342108940","76323368721600","248491290021975","494730748512000","586719747063450","379480971870000","102776096548125"],["1","65502","62702766","9670314680","490247463615","11170580090670","134976766904980","951083160387360","4125424986057375","11297640452348250","19514808978414750","20555219309625000","12024803296130625","2988412653476250"],["1","131036","189778599","40186125104","2680907291725","78789419670780","1218516580556275","11002366850995520","61738101649539315","222987737485012500","525348011202139125","800173577285082000","757562607656229375","404431845770452500","92854250304440625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","66","786","11348","192788","3769328","83383808","2059259808","56163653184","1676677413568","54382582971328","1904328098775040","71604040163981824","2877398525487444480"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","2"]]},"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":186497,"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-9062c0efbdc7f4fe"]},{"id":186498,"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-9062c0efbdc7f4fe"]},{"id":186499,"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-9062c0efbdc7f4fe"]}],"region_exact":{},"incoming_edges":[{"id":"e-9062c0efbdc7f4fe","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-8fb868076fadaac9","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":"055fc213526610b7e0dfb4e7852e089d4902ef9b2a2eee54d964390da22f6137","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-3fea9ad1525b48eb","role":"sequence","name":"Triangle of Ward numbers T(n,k) read by rows.","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-875fb560aee86673","kind":"fingerprint_related","node_a_id":"n-0490fcbf3cc940e2","node_b_id":"n-8fb868076fadaac9","fingerprint":"f6-5eb99b00596b7bfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5eb99b00596b7bfc"},"other_id":"n-0490fcbf3cc940e2","other":{"id":"n-0490fcbf3cc940e2","name":"derivative of Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","first_index":1,"fingerprint":"f6-5eb99b00596b7bfc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","6"]],"card_rows":[["1"],["1","6"],["1","20","45"],["1","50","315","420"]],"row_sums_preview":["0","1","7","66","786","11348","192788","3769328"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eda3c556a6b4857f","steps":["D"],"edges":["e-7674d32d5cd4f2d1"],"text":"A269939 \u2192 D"},"headline":"A269939 \u2192 D","chain":null}}],"oeis":[],"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,    6;\n   3: 1,   20,     45;\n   4: 1,   50,    315,      420;\n   5: 1,  112,   1470,     5040,      4725;\n   6: 1,  238,   5754,    37800,     86625,      62370;\n   7: 1,  492,  20475,   227920,    952875,    1621620,     945945;\n   8: 1, 1002,  68805,  1211980,   8183175,   24594570,   33108075,    16216200;\n   9: 1, 2024, 222948,  5948800,  60610550,  285044760,  662161500,   735134400,   310134825;\n  10: 1, 4070, 705276, 27659632, 407156750, 2801258460, 9955966020, 18745927200, 17677685025, 6547290750;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -n^2*x^2*P(n-2,x) - (n*x^2+n*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-n^2*x^2*P[n - 2, x] - (n*x^2 + n*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x)*P[n - 1, x] + (x + 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 + 6*x; for(n = 3, N, my(i = n); v[i] = -n^2*x^2*v[i - 2] - (n*x^2 + n*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3fea9ad1525b48eb","steps":["\u03b8+1"],"edges":["e-9062c0efbdc7f4fe"],"text":"A134991 \u2192 \u03b8+1"},"headline":"A134991 \u2192 \u03b8+1","chain":null}