{"id":"n-13d4582276f16445","revision":0,"first_index":0,"fingerprint":"f6-d8494e1c6027faf3","name":"theta_plus_one_squared of one_plus_derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","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":true,"rows":[[],["1"],["2","8"],["0","24","27"],["0","0","108","64"],["0","0","0","320","125"],["0","0","0","0","750","216"],["0","0","0","0","0","1512","343"],["0","0","0","0","0","0","2744","512"],["0","0","0","0","0","0","0","4608","729"],["0","0","0","0","0","0","0","0","7290","1000"],["0","0","0","0","0","0","0","0","0","11000","1331"],["0","0","0","0","0","0","0","0","0","0","15972","1728"],["0","0","0","0","0","0","0","0","0","0","0","22464","2197"],["0","0","0","0","0","0","0","0","0","0","0","0","30758","2744"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","10","51","172","445","966","1855","3256","5337","8290","12331","17700","24661","33502"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["8","-1"],["-20","3"],["18","-3"],["-7","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","-1","0"],["0","3","0"],["0","-3","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171452,"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-308e99db8c790030"]},{"id":171453,"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-308e99db8c790030"]},{"id":171454,"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-308e99db8c790030"]}],"region_exact":{},"incoming_edges":[{"id":"e-308e99db8c790030","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-13d4582276f16445","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":"f0a8781d7c2f09c3486d235b4bdd679408ccdd9d406e199d28649ecd1761beac","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-d16445f4fa1fddba","role":"sequence","name":"one_plus_derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","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-0f7e983e42e9672f","kind":"fingerprint_related","node_a_id":"n-13d4582276f16445","node_b_id":"n-309d1371dfdb9af6","fingerprint":"f6-d8494e1c6027faf3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d8494e1c6027faf3"},"other_id":"n-309d1371dfdb9af6","other":{"id":"n-309d1371dfdb9af6","name":"theta_plus_one_squared of one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-d8494e1c6027faf3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["0","72","81"]],"card_rows":[["1"],["4","16"],["0","72","81"],["0","0","432","256"]],"row_sums_preview":["1","20","153","688","2225","5796","12985","26048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D+1","(\u03b8+1)\u00b2"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-fd7b5be54bec4045"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-e37b397ba77aebf3","kind":"fingerprint_related","node_a_id":"n-13d4582276f16445","node_b_id":"n-637622bfa131edad","fingerprint":"f6-d8494e1c6027faf3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d8494e1c6027faf3"},"other_id":"n-637622bfa131edad","other":{"id":"n-637622bfa131edad","name":"theta_plus_one_squared of one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-d8494e1c6027faf3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["0","56","63"]],"card_rows":[["1"],["4","16"],["0","56","63"],["0","0","270","160"]],"row_sums_preview":["1","20","119","430","1157","2576","5035","8954"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-133ff869421f85a2","e-33ae1001ed87b48b"],"text":"A131431 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A131431 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","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: 0;\n  1: 1;\n  2: 2,  8;\n  3: 0, 24,  27;\n  4: 0,  0, 108,  64;\n  5: 0,  0,   0, 320, 125;\n  6: 0,  0,   0,   0, 750,  216;\n  7: 0,  0,   0,   0,   0, 1512,  343;\n  8: 0,  0,   0,   0,   0,    0, 2744,  512;\n  9: 0,  0,   0,   0,   0,    0,    0, 4608, 729;\n","recurrence_sentence":"The row polynomials satisfy (n^4-7*n^3+18*n^2-20*n+8+(n^3-3*n^2+3*n-1)*x)*P(n,x) = ((n^4-3*n^3+3*n^2-n)*x+n^3*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^4 - 3*n^3 + 3*n^2 - n)*x + n^3*x^2)*P[n - 1, x])/(n^4 - 7*n^3 + 18*n^2 - 20*n + 8 + (n^3 - 3*n^2 + 3*n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (((n^4 - 3*n^3 + 3*n^2 - n)*x + n^3*x^2)*v[i - 1])/(n^4 - 7*n^3 + 18*n^2 - 20*n + 8 + (n^3 - 3*n^2 + 3*n - 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-1c11ffc3c4cd47ed","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-9e777fb0c714fbf5","e-308e99db8c790030"],"text":"A132440 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A132440 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}
