{"id":"n-f4310eecc20ba49e","revision":0,"first_index":0,"fingerprint":"f6-312ef1e8ce9b56f4","name":"one_plus_derivative of basis_map_A007318 of Even-numbered rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["8","6","1"],["48","80","48","12","1"],["256","672","720","400","120","18","1"],["1280","4608","7168","6272","3360","1120","224","24","1"],["6144","28160","57600","69120","53760","28224","10080","2400","360","30","1"],["28672","159744","405504","619520","633600","456192","236544","88704","23760","4400","528","36","1"],["131072","860160","2609152","4845568","6150144","5637632","3843840","1976832","768768","224224","48048","7280","728","42","1"],["589824","4456448","15728640","34406400","52183040","58146816","49201152","32215040","16473600","6589440","2050048","489216","87360","11200","960","48","1"],["2621440","22413312","90243072","227278848","401080320","526417920","532267008","423641088","268849152","136913920","56010240","18330624","4752384","959616","146880","16320","1224","54","1"],["11534336","110100480","498073600","1419509760","2857697280","4318298112","5080350720","4762828800","3611811840","2235883520","1135140864","472975360","161241600","44651520","9922560","1736448","232560","22800","1520","60","1"],["50331648","530579456","2664431616","8477736960","19175833600","32790675456","44008538112","47501279232","41912893440","30561484800","18540634112","9390710784","3972993024","1400734720","409305600","98233344","19100928","2949408","351120","30800","1848","66","1"],["218103808","2516582400","13891534848","48813309952","122563854336","233985540096","352835338240","430963163136","433798447104","364176474112","257065746432","153363087360","77533560832","33228668928","12049956864","3681931264","941402880","199355904","34456576","4760448","510048","40480","2208","72","1"],["939524096","11777605632","70883737600","272629760000","752458137600","1586432573440","2655550177280","3621204787200","4095410176000","3890639667200","3132988784640","2151951892480","1265854054400","639012864000","276905574400","102850641920","32635299840","8798732800","1999712000","378892800","58938880","7367360","717600","52000","2600","78","1"],["4026531840","54492397568","355140108288","1483978309632","4465675468800","10305404928000","18961945067520","28555786321920","35849927393280","38022650265600","34401445478400","26739305349120","17943481221120","10428689940480","5258162995200","2300446310400","872252559360","285864284160","80628387840","19449830400","3978374400","682007040","96445440","11007360","982800","65520","3024","84","1"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","15","189","2187","24057","255879","2657205","27103491","272629233","2711943423","26732013741","261508830075","2541865828329","24571369673847","236393522034597"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","1"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["8","12","6","1"],["8","8","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184401,"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-b8d106062dd0f914"]},{"id":184402,"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-b8d106062dd0f914"]},{"id":184403,"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-b8d106062dd0f914"]}],"region_exact":{},"incoming_edges":[{"id":"e-b8d106062dd0f914","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f4310eecc20ba49e","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"fe01deb6323322a1ed4af5055d797fe0a0197905f6e02e0f689fb459d528d873","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-8c0973a753fcbe28","role":"sequence","name":"basis_map_A007318 of Even-numbered rows of Pascal's triangle.","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-eb4e7c4788b6d4c4","kind":"fingerprint_related","node_a_id":"n-f4310eecc20ba49e","node_b_id":"n-8f7f00b848049200","fingerprint":"f6-312ef1e8ce9b56f4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-312ef1e8ce9b56f4"},"other_id":"n-8f7f00b848049200","other":{"id":"n-8f7f00b848049200","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-312ef1e8ce9b56f4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","12","2"],["192","320","192","48","4"]],"card_rows":[["1"],["16","12","2"],["192","320","192","48","4"],["2048","5376","5760","3200","960","144","8"]],"row_sums_preview":["1","30","756","17496","384912","8188128","170061120","3469246848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["x \u21a6 x+1","D+1"],"edges":["e-9d7e9cd87d9b797c","e-d5826c1d9bc802e9"],"text":"A139548 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A139548 \u2192 x \u21a6 x+1 \u2192 D+1","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:       8,        6,        1;\n  2:      48,       80,       48,        12,         1;\n  3:     256,      672,      720,       400,       120,        18,         1;\n  4:    1280,     4608,     7168,      6272,      3360,      1120,       224,        24,         1;\n  5:    6144,    28160,    57600,     69120,     53760,     28224,     10080,      2400,       360,        30,        1;\n  6:   28672,   159744,   405504,    619520,    633600,    456192,    236544,     88704,     23760,      4400,      528,       36,       1;\n  7:  131072,   860160,  2609152,   4845568,   6150144,   5637632,   3843840,   1976832,    768768,    224224,    48048,     7280,     728,     42,      1;\n  8:  589824,  4456448, 15728640,  34406400,  52183040,  58146816,  49201152,  32215040,  16473600,   6589440,  2050048,   489216,   87360,  11200,    960,    48,    1;\n  9: 2621440, 22413312, 90243072, 227278848, 401080320, 526417920, 532267008, 423641088, 268849152, 136913920, 56010240, 18330624, 4752384, 959616, 146880, 16320, 1224, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+x)*P(n,x) = (8*n+8+(8*n+12)*x+(2*n+6)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 + (8*n + 12)*x + (2*n + 6)*x^2 + x^3)*P[n - 1, x])/(2*n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 8 + (8*n + 12)*x + (2*n + 6)*x^2 + x^3)*v[i - 1])/(2*n + 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-08f49539099949d5","steps":["x \u21a6 x+1","D+1"],"edges":["e-cfe8f3b6680af0c3","e-b8d106062dd0f914"],"text":"A034870 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A034870 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}
