{"id":"n-a08be4d759a7daf2","revision":0,"first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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"],["8","4"],["80","80","16"],["1088","1472","576","64"],["18944","30720","16640","3584","256"],["403456","741376","491520","148480","20480","1024"],["10174464","20529152","15683584","5877760","1146880","110592","4096"],["296681472","644415488","547864576","239353856","58490880","8028160","573440","16384"],["9824108544","22664445952","20969947136","10294394880","2957574144","511967232","52297728","2883584","65536"],["364237553664","883943276544","876541378560","472116101120","153535119360","31450202112","4073717760","322437120","14155776","262144"],["14950009405440","37891124232192","39827093323776","23149643038720","8312276910080","1938495111168","298658562048","30104616960","1903165440","68157440","1048576"],["673020704194560","1771589251104768","1957448978006016","1213164666486784","472670399365120","122326385426432","21556705099776","2595364012032","209694228480","10842275840","322961408","4194304"],["32974686576967680","89750756910956544","103575891430342656","67827637380186112","28312518132760576","7986148357963776","1568278266773504","216800855851008","20994890661888","1391502295040","59978547200","1509949440","16777216"],["1746828955232501760","4898691598755299328","5874574149304713216","4036233642303815680","1787665896601288704","542358920815968256","116514306597584896","18011596110430208","2009233348362240","160080343334912","8867228418048","323666051072","6979321856","67108864"],["99493974660989583360","286629808004187291648","355710420627115474944","254980312141628702720","118927225527342202880","38418295719689453568","8906554038674259968","1510955185780817920","189038177854423040","17397233452318720","1161338218872832","54595745218560","1709933854720","31943819264","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","176","3200","70144","1806336","53526528","1795424256","67277684736","2786234204160","126399377571840","6234593317355520","332235165810032640","19023057577354199040","1164785198992824729600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["8","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","32","-16"],["32","-32","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","8"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190834,"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-500633f88eec282c"]},{"id":190835,"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-500633f88eec282c"]},{"id":190836,"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-500633f88eec282c"]},{"id":190837,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-500633f88eec282c"]}],"region_exact":{},"incoming_edges":[{"id":"e-500633f88eec282c","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-a08be4d759a7daf2","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":"7247e845fe212597086aec3393400130f72abd7a9deb408638aae7e68f1d6c10","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f4e656360cd62235","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-03535f54b19fba73","kind":"fingerprint_related","node_a_id":"n-9db4cace849774b0","node_b_id":"n-a08be4d759a7daf2","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-9db4cace849774b0","other":{"id":"n-9db4cace849774b0","name":"one_plus_derivative of basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["45","45","9"]],"card_rows":[["1"],["6","3"],["45","45","9"],["459","621","243","27"]],"row_sums_preview":["1","9","99","1350","22194","428652","9526572","239660208"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610","D+1"],"edges":["e-86f8b58a2aa07621","e-06ab765377c0e2de"],"text":"A225470 \u2192 basis A013610 \u2192 D+1"},"headline":"A225470 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-8c20f0e2af08bc7b","kind":"fingerprint_related","node_a_id":"n-7c0b893bdd20dbf2","node_b_id":"n-a08be4d759a7daf2","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-7c0b893bdd20dbf2","other":{"id":"n-7c0b893bdd20dbf2","name":"one_plus_derivative of basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["20","20","4"]],"card_rows":[["1"],["4","2"],["20","20","4"],["136","184","72","8"]],"row_sums_preview":["1","6","44","400","4384","56448","836352","14026752"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","D+1"],"edges":["e-03658933e548767a","e-c3de98ad74329879"],"text":"A028338 \u2192 basis A013609 \u2192 D+1"},"headline":"A028338 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-910a770ec7a842eb","kind":"fingerprint_related","node_a_id":"n-0254a5a8457ca2c9","node_b_id":"n-a08be4d759a7daf2","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-0254a5a8457ca2c9","other":{"id":"n-0254a5a8457ca2c9","name":"reversal of one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","23","17"]],"row_sums_preview":["1","3","11","50","274","1764","13068","109584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D+1","reverse"],"edges":["e-d9ba82797b30d77a","e-46075d03abc6b2c3"],"text":"A130534 \u2192 D+1 \u2192 reverse"},"headline":"A130534 \u2192 D+1 \u2192 reverse","chain":null}},{"id":"l-984e8b6cb205d6ec","kind":"fingerprint_related","node_a_id":"n-fb40e29cda26e7b9","node_b_id":"n-a08be4d759a7daf2","fingerprint":"f6-9dfcd5dbd8ed9f8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9dfcd5dbd8ed9f8f"},"other_id":"n-fb40e29cda26e7b9","other":{"id":"n-fb40e29cda26e7b9","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-9dfcd5dbd8ed9f8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["17","23","9","1"]],"row_sums_preview":["1","3","11","50","274","1764","13068","109584"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["D+1"],"edges":["e-d9ba82797b30d77a"],"text":"A130534 \u2192 D+1"},"headline":"A130534 \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,            4;\n  2:           80,           80,           16;\n  3:         1088,         1472,          576,           64;\n  4:        18944,        30720,        16640,         3584,          256;\n  5:       403456,       741376,       491520,       148480,        20480,        1024;\n  6:     10174464,     20529152,     15683584,      5877760,      1146880,      110592,       4096;\n  7:    296681472,    644415488,    547864576,    239353856,     58490880,     8028160,     573440,     16384;\n  8:   9824108544,  22664445952,  20969947136,  10294394880,   2957574144,   511967232,   52297728,   2883584,    65536;\n  9: 364237553664, 883943276544, 876541378560, 472116101120, 153535119360, 31450202112, 4073717760, 322437120, 14155776, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(32*n-32)*x+16*x^2)*P(n-2,x) + (8*n-4+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (32*n - 32)*x + 16*x^2)*P[n - 2, x] + (8*n - 4 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (32*n - 32)*x + 16*x^2)*v[i - 2] + (8*n - 4 + 8*x)*v[i - 1]); 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-9c5c208df8605708","steps":["basis A013611","D+1"],"edges":["e-8a4919ac2dd1c88f","e-500633f88eec282c"],"text":"A225471 \u2192 basis A013611 \u2192 D+1"},"headline":"A225471 \u2192 basis A013611 \u2192 D+1","chain":null}