{"id":"n-b3e61a65eb822470","revision":0,"first_index":0,"fingerprint":"f6-0e60f7e2ac8ceeab","name":"theta_plus_one of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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","2"],["1","10","3"],["1","42","42","4"],["1","170","441","120","5"],["1","682","4224","2508","275","6"],["1","2730","39039","45760","10010","546","7"],["1","10922","355446","784212","307450","31668","980","8"],["1","43690","3215397","13021320","8666515","1508988","84966","1632","9"],["1","174762","29004108","212628316","232939525","64723386","5923820","201552","2565","10"],["1","699050","261299115","3440725200","6089273520","2609569326","365777797","19669408","434511","3850","11"],["1","2796202","2352740610","55400002020","156532744500","101251623960","20967609600","1676873880","57323475","867790","5566","12"],["1","11184810","21178859793","889537019800","3982568615025","3832897755960","1145539765020","131282910720","6529684590","150471750","1628055","7800","13"],["1","44739242","190626515352","14260830796524","100676136650375","142763401552590","60603162534600","9711294588960","676597726350","22302380100","362513580","2899260","10647","14"],["1","178956970","1715705747031","228427461431520","2534729454755030","5260293819873690","3136112266006755","690783610875840","65729622246930","2982013261500","68396761290","812962800","4940208","14210","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","14","89","737","7696","98093","1490687","26542518","545598045","12787451789","338242587616","10009697903337","328904762907595","11920843347833790"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182888,"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-c0270e0f672d7360"]},{"id":182889,"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-c0270e0f672d7360"]},{"id":182890,"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-c0270e0f672d7360"]}],"region_exact":{},"incoming_edges":[{"id":"e-c0270e0f672d7360","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-b3e61a65eb822470","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":"de50380c69f4162af1353f5f0e250113d8b9cb7d9a4d0314d2b85781650ea5f1","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-c856ca922af03076","role":"sequence","name":"Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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-f70f1872fb6ce3c2","kind":"fingerprint_related","node_a_id":"n-3b3ead46f39c15a4","node_b_id":"n-b3e61a65eb822470","fingerprint":"f6-0e60f7e2ac8ceeab","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e60f7e2ac8ceeab"},"other_id":"n-3b3ead46f39c15a4","other":{"id":"n-3b3ead46f39c15a4","name":"derivative of Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","first_index":1,"fingerprint":"f6-0e60f7e2ac8ceeab","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","2"]],"card_rows":[["1"],["1","2"],["1","10","3"],["1","42","42","4"]],"row_sums_preview":["0","1","3","14","89","737","7696","98093"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0164479e47952770","steps":["D"],"edges":["e-5afa7ad1cf9291ca"],"text":"A269945 \u2192 D"},"headline":"A269945 \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,      2;\n  2: 1,     10,        3;\n  3: 1,     42,       42,         4;\n  4: 1,    170,      441,       120,         5;\n  5: 1,    682,     4224,      2508,       275,        6;\n  6: 1,   2730,    39039,     45760,     10010,      546,       7;\n  7: 1,  10922,   355446,    784212,    307450,    31668,     980,      8;\n  8: 1,  43690,  3215397,  13021320,   8666515,  1508988,   84966,   1632,    9;\n  9: 1, 174762, 29004108, 212628316, 232939525, 64723386, 5923820, 201552, 2565, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 2*x^2*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (1+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 2*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 2*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*x)*v[i - 1] + 3*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-c856ca922af03076","steps":["\u03b8+1"],"edges":["e-c0270e0f672d7360"],"text":"A036969 \u2192 \u03b8+1"},"headline":"A036969 \u2192 \u03b8+1","chain":null}
