{"id":"n-08fd024edc5659fd","revision":0,"first_index":0,"fingerprint":"f6-31dd68f9934466a8","name":"theta_plus_one_squared of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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","8"],["1","16","18"],["1","24","81","32"],["1","32","180","256","50"],["1","40","315","800","625","72"],["1","48","486","1792","2625","1296","98"],["1","56","693","3360","7350","7056","2401","128"],["1","64","936","5632","16500","24192","16464","4096","162"],["1","72","1215","8736","32175","64152","67914","34560","6561","200"],["1","80","1530","12800","56875","144144","210210","168960","66825","10000","242"],["1","88","1881","17952","93500","288288","539539","604032","382239","121000","14641","288"],["1","96","2268","24320","145350","528768","1212848","1757184","1563705","800800","207636","20736","338"],["1","104","2691","32032","216125","906984","2469012","4412928","5119686","3718000","1574573","340704","28561","392"],["1","112","3150","41216","309925","1474704","4653138","9922560","14284998","13613600","8236228","2935296","538265","38416","450"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","35","138","519","1853","6346","21045","68047","215586","671667","2063449","6264050","18821793","56052059"],"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","8"]],"recurrence":{"denominator":{"coeffs":[["-2","4","0"],["10","-36","0"],["-16","104","-8"],["8","-120","40"],["0","48","-64"],["0","0","32"]]},"terms":[{"coeff":{"coeffs":[["-2","-20","-60","-24"],["10","84","192","88"],["-16","-88","-76","-24"],["8","-24","-264","-200"],["0","48","176","64"],["0","0","32","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-10","-28","-16"],["0","4","32","76","48"],["0","-4","-16","4","16"],["0","0","-24","-128","-128"],["0","0","0","-16","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172996,"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-861beb70746a3612"]},{"id":172997,"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-861beb70746a3612"]},{"id":172998,"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-861beb70746a3612"]},{"id":172999,"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-861beb70746a3612"]}],"region_exact":{},"incoming_edges":[{"id":"e-861beb70746a3612","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-08fd024edc5659fd","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":"f14136b735a244978640f49abd95fcb47fd27cb59737dbcfeb7c58df65309752","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2496e2f555756362","role":"sequence","name":"Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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-0425d83bf7a5afac","kind":"fingerprint_related","node_a_id":"n-08fd024edc5659fd","node_b_id":"n-148975a3f35d4c6f","fingerprint":"f6-31dd68f9934466a8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-31dd68f9934466a8"},"other_id":"n-148975a3f35d4c6f","other":{"id":"n-148975a3f35d4c6f","name":"theta_plus_one_squared of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-31dd68f9934466a8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","16"],["8","128","144"]],"card_rows":[["1"],["2","16"],["8","128","144"],["32","768","2592","1024"]],"row_sums_preview":["1","18","280","4416","66432","948736","12996608","172400640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-11143fa0a471b62a","e-150370f45f4e033d"],"text":"A086645 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A086645 \u2192 basis A013611 \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: 1;\n  1: 1,  8;\n  2: 1, 16,   18;\n  3: 1, 24,   81,   32;\n  4: 1, 32,  180,  256,    50;\n  5: 1, 40,  315,  800,   625,    72;\n  6: 1, 48,  486, 1792,  2625,  1296,    98;\n  7: 1, 56,  693, 3360,  7350,  7056,  2401,   128;\n  8: 1, 64,  936, 5632, 16500, 24192, 16464,  4096,  162;\n  9: 1, 72, 1215, 8736, 32175, 64152, 67914, 34560, 6561, 200;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-16*n^2+10*n-2+(48*n^4-120*n^3+104*n^2-36*n+4)*x+(32*n^5-64*n^4+40*n^3-8*n^2)*x^2)*P(n,x) = (8*n^3-16*n^2+10*n-2+(48*n^4-24*n^3-88*n^2+84*n-20)*x+(32*n^5+176*n^4-264*n^3-76*n^2+192*n-60)*x^2+(96*n^5+64*n^4-200*n^3-24*n^2+88*n-24)*x^3)*P(n-1,x) - ((4*n^2-4*n+1)*x+(24*n^3+16*n^2-32*n+10)*x^2+(16*n^4+128*n^3-4*n^2-76*n+28)*x^3+(64*n^4+128*n^3-16*n^2-48*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 24*n^3 - 88*n^2 + 84*n - 20)*x + (32*n^5 + 176*n^4 - 264*n^3 - 76*n^2 + 192*n - 60)*x^2 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^3)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (24*n^3 + 16*n^2 - 32*n + 10)*x^2 + (16*n^4 + 128*n^3 - 4*n^2 - 76*n + 28)*x^3 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 120*n^3 + 104*n^2 - 36*n + 4)*x + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 24*n^3 - 88*n^2 + 84*n - 20)*x + (32*n^5 + 176*n^4 - 264*n^3 - 76*n^2 + 192*n - 60)*x^2 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^3)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (24*n^3 + 16*n^2 - 32*n + 10)*x^2 + (16*n^4 + 128*n^3 - 4*n^2 - 76*n + 28)*x^3 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 120*n^3 + 104*n^2 - 36*n + 4)*x + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^2)); 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-2496e2f555756362","steps":["(\u03b8+1)\u00b2"],"edges":["e-861beb70746a3612"],"text":"A084534 \u2192 (\u03b8+1)\u00b2"},"headline":"A084534 \u2192 (\u03b8+1)\u00b2","chain":null}